PSYCH RESEARCH I
PSYCH RESEARCH I PSYC 201
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Date Created: 10/19/15
Using Pearson s Correlation Coefficient I Correlation Coefficients Correlation coefficients in are a class of statistical tests that provide standardized estimates of the strength of the relationship between 2 variables The type of data nominal ordinal intervalratio determines the statistic that you use For example Cramer s Vaka 4 is used for discrete data In general they represent the ratio of the variance shared between two variables Covariance divided by the total variability in both measures Squaring a correlation coefficient tells us the of the variance in the Dependent Variable that is explained by predicted by accounted for by variability in the Independent Variable II Pearson s r Used with two continuous variables Estimates the Strength and Direction of the Relationship between the IV and DV Scores range from 1 to l the sign of the coefficient indicates the direction of the relationship 1 perfect negative correlation inverse relationship As one variable increase the other variable decreases l perfect positive correlation direct relationship As one variable increase the other variable increases the closer the absolute value of r is to l regardless of wether it is positive or negative the stronger the relationship between the two variables the closer the value of r is to 0 the weaker the relationship is between the two variables Statistical Hypotheses HO r 0 HA r 0 we are testing whether or not the strength of the relationship between the IV and the DV differs from what we would expect by chance alone Does it significantly differ from 0 Example Research Question What is the strength and direction of the association between the number of Vienna Sausages that a person eats in 24 hr period measured in grams range 0900 and the amount of time they spend doing impersonations of south park characters in that same 24 hr period measured in minutes range 01440 III Calculating Pearson s r Pearson s r Allows us to test the strength of the association between two continuous variables It represents a ratio of the Covariance variance shared by two variables and the total variance covariance unique variance df n 2 ZXXZ Y ZXY ZX XY Y n r 0 72f EXZ XXV ZYZ EYY 12 X X2Z Y Y2 n n The Coefficient of Determination r represents a ratio of covariance to total variance but if we want to know what percent of the variance in the DV is explained by variance in the IV then we can square r to find out lr2 error variance residual variance unexplained varianc the amount of variance as a percentage that is is attributed to sources other than the independent variable IV How Does r Work What the r statistic does is divide up the variance or the deviation in the data set The deviation of each observation subject score is made up of two parts Total Variance Variance Shared by Variables Variance due to Error Covariance Covariance the amount of variance in Y that is explained accounted for by the variance in X X Y ZX XY Y ZXY W Covariance Ky Or n 1 n 1 Variance due to Error the amount of variance in Y that is not explained shared with accounted for by the variance in X What the r formula does is determine the total amount of Covariance and then divide this value by the total amount of variance Pearson s r Formula EXXZ Y ZXY n r EXZ 2Y2 2X21 Hz n n r Covariance Total Variance for Each Variable V Signi cance of r Compare the absolute value of the r obtained with the r critical for df n 7 2 a the p 05 level The table below presents the critical values of Pearson s r Ifr obtained gt r criticaldfn27p osvtwomled then reject H0 and Fail to reject HA Ifr obtained lt r criticaldfn27p osytwomled then reject HA and Fail to reject Ho The table below presents One and Two tailed tests For our purposes we are really only interested in a Two Tailed test There are occasions where a One Tailed test is permissible but they are relatively rare and we don t need to worry about them at this point VI Report your results in APA format If signi cant rdf p lt 0 eg r66 42 p lt 001 Ifnonsignif1cant rdf p gt 05 eg r66 22p gt 05 VII Effect Size Cohen 1988 offers standards for evaluating the Effect Size of Pearson s r Effect Size 1 12 Small 10 01 Medium 30 09 Large 50 25 SPSS GUIDE 1 Psych Research 1 Guide to SPSS 110 I What is SPSS SPSS Statistical Package for the Social Sciences is a data management and analysis program It allows us to store and analyze very large amounts of research data The statistics that SPSS is capable of are far more complex that the stats that we can do in excel which makes it more desirable as an analysis tool Also spss allows us to store our data protocols syntax and results output in separate les which makes analysis of large amounts of data much less cumbersome than excel II Goals Our goals for this unit include the following aspects Learn to set up data les and enter import data III Learn to create syntax files VI Learn to generate descriptive statistics V Learn to generate Frequency statistics VI Learn to compute new variables from existing variables VII Learn to transform normalize variables VIII Learn to use lters selecting cases IX Learn to compute chi square single variable multivariate X Learn to compute correlation coef cients XI Learn to compute ttests independent sample repeated measure XII 5090839 59 III Data Files Set up Entering Data amp Importing Data For the most part SPSS can be split up into 3 major parts The Data Editor where we enter data name variables compute new variables and select cases The Syntax Editor where we store amp create syntax for our analyses and procedures and The Output Navigator where we view the results our statistical tests have generated The rst step in this process is to set up the data le in the Data Editor note we can create variables and enter data in the syntax editor but it is beyond the scope of this course 39f we have already created a data le in another program we can import it by selecting open on the File pulldown menua An open le box will appear If the desired le is not an spss le sav scroll through the file type options and choose the appropriate format eg excel xls Then open the desired le Note if desired le is not created in spss a second box will appear If you have variables named in this le then check the read variable names box other wise leave blank Also you can designate portions of les to open using the range option see help for details De ning Variables If you are creating a new data le editing an existing data le or have imported a data le where the variables have not yet been named andor have not had value labels associated with the variables then you will need to begin here there are 2 parts or views to the Data Editor 1 The Data View This view allows you to view and input data values The columns represent Variables and the rows represent participants subjects often referred to as cases 2 The Variable View This view allow you to edit variables and add new variables to the data stet Note that the rows represent each variable and corresponds to the columns in the data view Also the columns in this view represent different aspects of each variable 3 To toggle back and forth between the Data View and the Variable View Click on the small labeled le tabs located at the bottom left hand SPSS GUIDE corner of the data editor spread sheet 4 To create variables on a blank data le select the variable view Note that no variables are listed a Give the variable a Variable Name Variable Names are limited to 8 characters a throw back from the old Unix days The Name can not start with a number though it can have numbers in it Nor can it have any spaces or symbols ie Aamp gtlt in it except and 7 b Give the variable a Variable Label by clicking the appropriate cell in the Label column Variable Labels are are more exible than Variable names You can have more than 200 characters and you can use spaces and symbols The variable label allows you to give a more descriptive name to the variable that will make sense to you when you come back and look at your data after a long period of time has elapsed Be as precise as possible Also when you give a variable a Variable Label the variable label will appear on the output of your analyses see below However notice that labels more than 40 or so characters will not be truncated on the output it will only present the lSI 40 characters so put the most unique and descriptive information rst Otherwise your outputs may become vary confusing c If the variable is a categorical variable e g gender class rank ethnicity group membership then you will need to de ne the Value labels You do not need to do this if your data is ratio interval or ordinal data like 17 numerical rating scales By adding Value Labels you will not have to try to remember what the numbers stand for e g l male 2 female Also the Value Label will be printed on the output cl To de ne Value Labels click on the appropriate cell of the Values column A dialogue box labeled Value Labels will appear c2 In the value field rst type the lowest value e g 0 or 1 then in the value label eld type the name of the category e g female and nally click the add button c3 next in the value field type the next highest value eg l or 2 then in the value label eld type the name for that category eg male and nally click the add button c4 continue this process until all the values for all the categories have been named When this is complete leftclick OK It is always a good idea to make your rst variable subject number subnum so that if certain subjects have to be excluded from later analyses e g because of missing data or some other criterion they can be easily sorted out based on subject number Enterng DataOnce the variables are named and the values labeled you can begin data entry In the Data View simply select the cell you want to begin with type the appropriate value into that cell and then either press enter or one of the direction arrow keys If entering data on the righthand numerickeypad be sure that the number lock has been turned on and be careful not to turn it off when reaching for the 7 key Data can be corrected by selecting the desired cell and typing in the new value and pressing either enter or a directional arrow key SPSS GUIDE Saving Files70nce you have your variables de ned and perhaps some data entered you should periodically save your le If you have imported your data from another program DO NOT overwrite your original le make a new le and save it as an spss data le sav Chose save as from the le pull down menu and save the le in the appropriate directory e g a Note you do not have to give the le a name different from the original data le because it will have a different extension ie sav not xls if the original le was from excel and will therefore not overwrite your original data le Be sure to save your work often and save it in multiple places eg make backups so you will not loose anything important IV Smtax Files What is SyntaxIn the olden days 810 years ago there did not exist a Graphical User Interface GUI version of spss and all data entry and analysis was done using an spss syntax language Now we have a windows based program a GUI and we can point amp click our way through all analyses The only problem with this is that when we want to change our analysis procedure we must step through the whole point click process again which is tiresome and potentially error ridden However throughout spss there is the option to paste syntax which will send our point click commands to a syntax sheet which can be stored as a separate le This way we can have all the steps in our analysis recorded and stored Also we can run all of our analysis from the syntax sheet write syntax for new analyses or alter the syntax from previous analyses Opening a Syntax Sheet if no syntax sheet is open when we paste our rst procedure then spss automatically opens an untitled sheet for us Later we can title and save as a syntax le sps Otherwise it will paste syntax to the bottom of our currently opened syntax sheet To create a new syntax sheet select new from the file pulldown menu and then select syntax from the side menu To open an existing syntax le select open from the le pulldown menu The open le dialogue box will appear You will need to change the le type line to sps Otherwise the desired le will not appear as an option only sav les will Finally choose the desired le from the desired directory and click open Editing Syntax Syntax can be moved and removed using the copy cut amp paste commands identical to those found in most word processors and spreadsheet programs Running Analyses From Syntax to run an analysis or multiple analyses from the syntax box highlight the syntax for the desired analyses and then press the play button on the tool bar The play button gt resembles the play button on a tape recorder or VCR arrow pointing to the right Also analyses can be run from the run pull down menu where there are four options runs all analyses on all syntax sheets that are open selection runs highlighted area current runs all analyses on the currently active syntax sheet and to end runs analyses on the current syntax sheet that fall below the point where the cursor is currently positioned V Output Files Anytime you run an analysis eg Desriptives Frequencies ChiSquare etc the results will be presented in a separate window titled Output Viewer Like syntax this is a separate le that will need to be save The extension for these types of les are spo short for spps output Page Setup and Saving When you generate output that you plan on saving and printing out then you should select the Page Setup option from the pulldown File menu A Page Setup dialogue box will appear Click on the Options button A second Dialogue box will appear titled Page LA SPSS GUIDE Setup Options Here you can give your le a header that will always appear on the printed output Clicking on the Calendar icon will give the date the output was printed The clock gives the time it was printed gives you the page number very handy if you accidentally mix the pages The icon that looks like a sheet of paper will print the le name on the header Also you can type your own message in the header This makes keeping track of your output very easy 7 To save the le simply click the oppy disc icon or select save or save as from the pull down le menu Opening an Output File To open an existing output le select open from the file pulldown menu The open le dialogue box will appear You will need to change the le type line to spo Otherwise the desired le will not appear as an option Finally choose the desired le from the desired directory and click open Note that all analyses can be run from the output window but data can not be entered and Transformations of the data See below can t be made from this window V Descriptive Statistics What are Descriptive Statistics 7 This procedure will give you a variety of basic statistical options Mean Sum Standard Deviation Variance Range Minimum Maximum Standard Error of the Mean Kurtosis Skewness 7This type of data is the rst analysis of any variable It allows us to determine if a measure is behaving in the way we want e g does it have enough variance to be useful is it skewed is it kurtotic is the range adequate or is there some kind of ceiling or oor effect Generating Descriptive Analyses 7 in either the Data Editor Syntax Sheet or Output Navigator select Descriptive Statistics from the Analyze pulldown menu and then select the Descriptives option from the side menu A descriptives box will appear In the left hand column appear all the variables you have data for Move the variables you want analyzed to the variables column Do this by either doubleleftclicking on the desired variable or singleleftclicking on the desired variable and then leftclicking on the boxed arrow between the columns To remove a variable from the variables list follow the same prodedures 7 To choose the descriptive statistics you want left click the options button the default stats are Mean Standard Deviation Minimum and Maximum and check the boxes of the stats you want In general it is best to ask for all the stats possible just in case you have a need for them later Then click continue 7To paste the syntax for the descriptive stats you want leftclick the paste button in the Descriptives box 7To run analyses without pasting them to the syntax sheet leftclick OK Interpreting Descriptive Analyses Output Descriptive output has 2 components Title Descriptive Statistics Each can be displayed or hidden by doubleleft clicking on their label in the outline located in the lefthand column Within the Descriptive Statistics box the rows represent variables and the columns indicate the various requested statistics The last row is labeled Valid N Listwise this refers to the number of subjects who are not missing any data for any of the variables listed The casewise N s are listed above this these are the number of subjects who are not missing the data for each variable each case VI Frequency Statistics What are Frequency Statistics The Frequency Analysis allows us to generate ungrouped 4 SPSS GUIDE 5 frequency tables tells us how often each score occurred for our variables of interest while also generating all of the statistics that the Descriptives command allows us to generate although we can not get a listwise valid N The ungrouped frequency tables are useful for providing us with a visual representation of the distributions for our variables something we can t get from descriptive statistics alone Further within the Frequencies command we can generate graphical displays of our data eg histograms bar charts pie charts line graphs scattergrams etc Generating Frequencies 7 in either the Data Editor Syntax Sheet or Output Navigator select Descriptive Statistics from the Analyze pulldown menu and then select the Frequencies option from the side menu A Frequencies box will appear In the left hand column appear all the variables you have defined Move the variables you want analyzed to the variables column by either doubleleftclicking on the desired variable or singleleftclicking on the desired variable and then leftclicking on the arrow between the columns To remove a variable from the variables list follow the same procedures 7 Statistics To choose the descriptive statistics you want the default stats are None left click the Statistics button and a Frequencies Statistics box will appear This box is comprised of four parts Percentile Values Central Tendency Dispersion and Distribution Percentile Values has three options Quartiles this will generate the cut off points for dividing your subjects into 4 equal groups for the selected variables Cut Point For Equal Groups generates cut off points for any number of equal sized groups you select Percentile will give you the raw scores associated with any percentiles you Add to the percentile list Central Tendency contains the following options Mean Median Mode and Sum Dispersion contains the options Standard Deviation Variance Range Minimum Maximum and Standard Error of the Mean SEM Distribution contains options for Skewness and Kurtosis which tell us how far our distribution deviates from a normal distribution Check the boxes of the stats you want Again it is best to ask for all the stats possible just in case you have a need for them later but the percentiles are not that necessary unless you know you will use them for something Then click Continue iCharts To generate graphical representations e g figures of your data distributions then click on the Charts button There are three main chart types available on in the Charts box Bar Chart for use with discretecategorical data Pie Chart for use with both discretecategorical amp continuous data Histogram for use with continuous data note more chart options are accessible from the Graphs pulldown menue on the uppermost tool bar For the Bar chart and the Histogram there is also the an option that allows you to display the estimated normal curve for your data This option allows you to see how much your true distribution deviates differs from the theoretical normal distribution e g how much skewness and kurtosis your data has For all charts you also have the option of representing either the absolute frequency ie the total number of occurrences of each score or the percentage ie the total number of occurrences of each score divided by the total number of scores iFormati to specify how you want the frequency distribution to appear leftclick the Format button There are 5 options in the Frequencies Format box Ascending Value Will order the chart so that the smallest scores are at the top and largest are at the bottom Descending Value will order the chart so that the largest scores are at the top and the smallest scores are at the bottom Ascending Counts will order the chart so that the least frequent scores are at the top and the most frequent scores are at the bottom SPSS GUIDE 6 Descending Counts will order the chart so that the most frequent scores are at the top and the least frequent scores are at the bottom Suppress tables with more than Categories this allows you to exclude tables for variables where all scores have a frequency of 1 In such a case if you have many subjects your table will be very large and not particularly informative This option will allow you to still get the stats you want and the valid number of subjects and missing cases for that variable but with out the frequency table iSyntax To paste the syntax for the descriptive stats you want leftclick the paste button in the Frequencies box 7To run analyses without pasting them to the syntax sheet then leftclick OK Interpreting Frequency Output Frequencies output is made up of 2 or 3 major parts depending on whether you asked for charts 1 Part one is labeled Statistics Statistics will give you at least the Valid N and Missing N Valid Missing total N Depending on what statistics you asked for various stats will follow the data for N 2 Part two is labeled with the Variable Label The Rows of this table are labeled with the Value Labels you defined for your variable if it discrete if it continuous the rows will have the scores The Columns are labeled with the following headings Frequency Percent Valid Percent Cumulative Percent Frequency this is the Absolute Frequency eg the number of participants in that category or with that score Percent this is the Relative Frequency with respect to the Total N Valid Percent This is the Relative Frequency with respect to the Valid N number of participants with data for that variable Cumulative Percent This Cumulative Relative Frequency CRF tell you the percent of observations participants that have accumulated up to the score of interest eg it includes all the scores above it on the table For example the CRF for the second row is the sum of the CRF for the first row plus the RF for the second row Similarly the CRF for the 3rd row is the sum of the CRF of the 2 1d row plus the RF for the 3rd row 3 Part 3 will only be included if you requested Charts This section will also be Titled with the Variable Label Charts can be edited by doubleleft clicking on the chart itself A new window will open called Charts Editor here you can make alterations to your charts Several useful options can be accessed by selecting the Charts pulldown menu Any changes you make to the chart in the charts editor will be re ected in your output To return to output either close the chart editor or use the windows pulldown menu and select the file you want to return to e g the name of the output file you are working on VII Compute Commands What are Compute Statements 7 compute statements allow us to create new variables by transforming existing variables or to alter an existing variable by using some mathematical function It may be as simple as subtracting 1 from every participant s score or more complex like computing an interaction score between several variables The most frequent use of compute statement is to create a scale score by averaging several items from a questionnaire Writing Compute Statements 7 From the Transform pulldown menu select compute and the compute box will appear In the upper left hand comer is the Target Variable field where you designate the name of the variable that you want to create if the name you specify already exists the program will ask if you want to overwrite copy over the existing variable before it executes the compute command To the right of the target variable field is the Numeric Expression field This is where you SPSS GUIDE 7 write the formula for the new variable For example if you want to reverse the direction of scores for a questionnaire item using a 7 pt likert scale with a variable name of ql the numeric expression would be 8ql The operators for these equations are as follows addition lt greater than 7 not subtraction gt less than 7 not equal to multiplication lt greater than or equal to amp and division gt less than or equal to l or exponentiation equal to Grouping operator eg order of operations 7 Functions The functions list provides some useful shortcuts for certain operations Note brief descriptions of the functions can be accessed by rightclicking on the function in question A small dialogue box will appear explaining what the function does Meannumexpr numexpr Will average the variables numexpr that you include in the parentheses Sumnumexpr numexpr Will sum the variables that you include in the parentheses LGlOnumexpr Returns the base 10 log of a variable LNnumexpr Returns the natural log base e of a variable SQRTnumexpr Returns the square root of a variable ABSnumexpr Returns the absolute value of a variable 7 Conditional Compute Statements the IF button allows you to make the compute statement conditional That is you can have SPSS preform the compute statement only for cases subjects were certain conditions are met For example if you only wanted the data for male participants to be transformed you could include the IF condition gender l 7 Syntax to paste the syntax for the descriptive stats you want leftclick the paste button in the compute box 7To run a compute statement without pasting it to the syntax sheet leftclick OK VIII Transformations Normalizing Variables Why Normalize Variables 7 when a variable is found to significantly deviate from the normal distribution ie it has a skewness or kurtosis with an absolute value greater than 100 then we need to transform the data so that it has a more normal distribution This can be done using the compute statement see compute statements above appropriate to the type of problem you have For the most part you can follow these rules If positively skewed skewed right use a logarithmic function of some sort e g LGlO 7If negatively skewed skewed left use an exponential function of some sort e g square it or cube it IX Filters Selecting Cases Creating Subsets What are Filters Filters are kind of what they sound like they allow you to exclude cases participants that you do not want included in an analysis For example if you wanted to know the mean IQ of only your male participants you could apply a filter that excludes the female participants and then generate the descriptive or frequency statistics for the IQ variable Also filters can be used to exclude certain participants based on their subject number For example if subjects 10 and 25 did not complete a questionnaire you could apply a filter that excludes all participants with subject SPSS GUIDE 8 numbers equal to 10 amp 25 there would be one of each unless you accidentally repeated a subjected number in your data entry Selecting Cases the select cases option makes use of lters to exclude unselected cases That is unless cases satisfy some condition that you specify then the case will be filtered out To select cases choose the select cases option from the Data pulldown menu A select cases box will appear with several options The default option is All Cases which mean all participants are included an no cases are excluded To select a subset of cases check the If condition is satis ed circle The IF box will now be active and you should leftclick on it A Select Cases If box will open The upperrighthand field is where you write the conditional statement that you want satisfied Examples 7 Males only female l amp male 2 Gender 2 7All subject except 10 amp 25 Subnum N 10 amp Subnum N 25 7Subjects 100 and up Subnum gt 100 Also subnum gt 99 7 if scale score grater than 4 Meanql q2 q3 gt 4 Once the conditional statement is specified then leftclick continue and you will return to previous select cases box 7Before continuing on be sure that the Filtered option in the Unselected Cases Are field located at the bottom of the dialogue box is select an not the Deleted option the deleted option will delete the unselected cases from your data file 7Syntax7to paste the select cases command syntax to the syntax sheet leftclick the paste button Note the filter will not be applied until you run the syntax or return to the select cases option and click OK 7 to apply the filter without pasting to the syntax sheet then just leftclick OK Note If you are not sure whether filters have been activated you can check by viewing the data in the data editor If a filter has been applied then you will see the appropriate row markers left hand side of spreadsheet will have diagonal slashes through them If no row markers have slashes through them then you have not activated any filters Also be sure to deactivate filters by returning to select cases and selecting All Cases There nothing more frustrating than running a bunch of analyses and discovering that you have only included half of the subjects because you left a filter on X Chi Sguare What is a Chi Square A chi square is one type of nonparametric distribution free test A chi square allows you test whether the frequency of occurrence for different categories of a discrete categorical variable or variables are significantly different from the frequencies expected by chance alone That is it tells us if there is some systematic difference in the number of people we have in each category or whether the differences can be explained by random chance The are two types of chi squares we will focus on here Univariate one discrete variable and Bivariate two discrete variables Univariate Chi Square 7From Analyze pulldown menu select Nonparametric Tests and then select Chi Square from the side menu A chi square box will appear From this command box you can request several separate chisquare tests at once Enter all variables that you want to be analyzed in the Test Variable List by either doubleleftclicking on the desired variable or by selecting the variable and then leftclicking on the boxed arrow 7The Expected Range option allows you to remove certain groups from the test For example if you have 6 groups but only want to test 4 of them then you can set your range SPSS GUIDE 9 from 2 to 5 excluding l amp 6 7The Expected Values option allows you to customize your expected frequencies if you have a theoretical reason for doing so e g gender distributions not really 5050 femalemale but rather 5644 so if you are testing gender you may want to customize your expected frequencies to re ect this natural pattern 7The Options button opens a box where you can choose between some statistics descriptives and quartiles and treatment of cases with missing data Excluding cases test by test means that each chisquare test will include all participants that have data for that particular variable being tested Excluding cases listwise means that each chisquare test will only include participants who have data for all the different variables listed in the Test Variable List 7Syntax to paste syntax to the syntax sheet leftclick the Paste button in the Chi Square Test box 7to run the chisquare analysis without pasting to the syntax sheet leftclick OK Interpreting a Univariate Chi Square Output Descriptive Statistics 7 these will only be presented if you selected descriptives from the options box They present the Nnumber of cases analyzed mean and standard deviation which is rather meaningless for categorical data and the minimum and maximum values 7 Contingency Table7 This table will be labeled with the variable label that you specified when you crated the variable The first column will present either the values numbers of the categories or the value labels names of the categories if you identified them when you created the variable Column 2 shows the Observed Frequencies for each category Column 3 shows the expected frequencies for each category Column 4 shows the Residual Observed Expected 7Test Statistics This box gives you the value of ChiSquare statistic the degrees of freedom number of groups l and the Significance level achieved anything less than 05 means that the observed frequency is significantly different from the expected frequency Bivariate Chi Square 7 From the Analyze pulldown menu select Descriptive Statistics and then select Cross Tabs from the side menu From the Cross Tabs box you can preform a variety of multivariate nonparametric tests To designate a ChiSguare test leftclick on the Statistics button From the statistics box select the chisquare option upper lefthand comer and then click continue Enter the variable with the fewest number of categories in the Columnsgs box by leftclicking the desired variable and then leftclicking the boxed arrow pointing to the Columnss box Enter the variable with the most categories in the Rowgs box following the same procedure described above 7Cells7 the Cells button gives you options regarding the information that will be displayed in the contingency table You can ask for both the Observed cell frequencies and the Expected cell frequencies Also you can also ask for the Row Column and Total percentage to be included in each cell The Residuals Observed Frq Expected Frq can be included as unstandardized standardized and adjusted standardized scores Residuals are standardized by dividing them by a mean error estimate and they have a mean of 0 and standard deviation of 1 Adjusted Standardized Residuals are reported in standard deviation units from the mean Note It is generally a good idea to ask for all of the options in the cells box Although it does clutter the chart some this information can give you better idea of what is happening with your data SPSS GUIDE 10 iFormati this gives you the option of determining whether your groups will be but in ascending or descending order Either is ne iSyntax to paste syntax to the syntax sheet leftclick the Paste button in the Cross Tabs box ito run the chisquare analysis without pasting to the syntax sheet leftclick OK Interpreting a Bivariate Chi Square Output Case Processing Summaryi gives you a breakdown of the number of valid cases individuals with all data that were included in the analysis number of missing cases individuals missing one or more data points and were excluded 7C0ntingency Table This table will be labeled with the variable labels that you speci ed when you crated the variables The columns and rows will be labeled with either the values numbers of the categories or the value labels names of the categories if you identi ed them when you created the variable The Observed Frequencies Expected Frequencies Residuals Observed Expected and requested percentages are displayed in each cell iChi Square Tests The results of the ChiSquare can be obtained from the rst row ofthis table labeled Pearson ChiSquare The test statistic is located in column 2 The degrees of freedom are shown in column 3 The signi cance level is shown in column 4 A chisquare statistic is signi cant if the signi cance value is less than 05 XI Correlations What are Correlations A correlation allows you to test the direction magnitude and signi cance of the association between two continuous variables This procedure produces the 2 statistic also called Pearson s Product Moment Correlation Coef cient which has an absolute value ranging between 0 and l A positive correlation indicates a positive relationship between variables eg as one increases the other increases A negative correlation indicates a negative relationship eg as one variable increases the other variable decreases What a correlation tells us is how much variance in a variable is shared by the variance in another variable this is called covariance The E statistic itself is a ratio of the Covariance divided by the Total Variance By squaring the 2 statistic we get an estimate of the amount of variance in one variable that is accounted for by another variable Coef cient of Determination Generating Correlations From the Analyze pulldown menu select Correlate then select Bivariate from the side menu In the Bivariate Correlations dialogue box enter the variables to be tested in the Variables eld by either doubleleft clicking on the desired variable or left click on the desired variable and then leftclick the boxed arrow You must enter at least two variables in the Variables field in order to preform a test After listing your test variables select the desired correlation coef cient in this case we want the Pearson option You must also select the desired test of signi cance In general we will always want the Two Tailed option Also check the box to Flag Signi cant Correlations This will make it easier to identify signi cant associations on the output tables iOptionsi leftclicking on the Options button will open the Options dialogue box with options for stats and treatment of missing data In the stats options you can ask to include means and standard deviations and the cross products and covariances of the variables these are not necessary In the missing values options you can ask to exclude cases pairwise this will exclude participants only in analyses where they do not have values for each variable being tested or listwise this will exclude participants for a all tests if they are missing a data point for any of the variables you have put in the Variables list iSyntax to paste syntax to the syntax sheet leftclick the Paste button in the Bivariate SPSS GUIDE 11 Correlations box J With Syntaxi one option that is available from the syntax sheet that is not available from the dialogue box it the ability to include the With command By inserting with between variables in the syntax as follows CORRELATIONS VARIABLESheight with pantsize weight PRINTTWOTAIL NOSIG STATISTICS DESCRIPTIVES XPROD MISSINGPAIRWISE You can tell the computer to only generate 2 correlations height with pantsize amp height with weight instead of the standard 3X3 matrix with 9 correlations height x height height x pantsize height x weight pantsize x height pantsize x pantsize pantsize x weight weight x height weight x pantsize weight x weight This can make interpretation of your output much easier ito run the chisquare analysis without pasting to the syntax sheet leftclick OK Interpreting Correlation Output Output for Correlations produces a table called a Correlation Matrix The format of this matrix will depend on the stats you requested and on the syntax you used to produce it ie whether you used the wit command or not 7Standard Matrix no with the rows of this matrix will be made up of 3 major sections Pearson Correlations Signi cance and N unless you asked for stats option 2 in which case the matrix will include a section for sums of squares and crossproducts and a section for covariance The columns of the matrix will be titled with the labels not the names of the variables you selected iPearson correlationsi this section will have X2 number of correlations where X number of variables requested In the left to right diagonal you will see a series of 100 these are the correlations for each variable with itself which is a perfect positive correlation You will also notice that the correlations above e g to the right of the diagonal are identical to the correlations below the diagonal this is because they are the same tests only with order of the variables in the equation reversed Correlations with asterisks by them are significant at least at the p lt 05 level eg 95 con dence level isigyiflcanc 7 more specific significance levels are reported here The maximum significance value reported is p lt 000 this means it is at least significant at the plt0001 level Note df n2 J With Syntax Matrix This matrix differs from the standard matrix in that it will not include the correlations of the variables with themselves or the correlations between the variables that appear on the same side of the with statement in the syntax For example with the syntax Variables AGE IQ With SHOESIZE BEDWETNG you will get the following correlations Age x shoesize age x bedwetng iq x shoesize iq x bedwetng But you will not get age x age age x iq iq x iq shoesize x shoesize shoesize x bedwetng or bedwetng x bedwetng XI t Tests What is a t Test 7 tTests allow us to determine whether two groups have significantly different averages for some continuous dependent variable The independent variable in this case is a discrete variable made up of group membership e g test group vs control group For example assume that you wanted to see if drinking beer makes you burp more than not drinking beer so you randomly assign your participants to one of two groups Beer grp Vs No Beer grp and then you count the number of belches that occur during a one hour test period iThree Different Types of t Testy SPSS GUIDE 12 Dependent Sample tTest the One Sample T Test7 For this type of test we are comparing a sample mean with the population mean These types of tests are rare because it requires that we know a population parameter One example of the use for this test would be to compare the number of children a sample of Midwestern families have to the number of children all US families have Population Parameter obtained from Census Bureau Independent Sample tTest 7 For this type of test we have two sample groups for which we have averages on some continuous variable An example of this can be seen in the Beer and Burping example given above Repeated Measures ttest Paired Sample T Test 7 For this type of test we again only have one sample group but we have data drawn from two different time points For example if I wanted to know if taking statistics improved your overall math ability I could test your math ability on the first day of class and then again on the last day of class and compare the class before and after averages iGenerating a t Testi 70ne Sample T Test From the Analyze pulldown menu select Compare Means and then select One Sample T Test from the side menu This test requires you to identify one continuous variable in the Test Variables Field by doubleleftclicking on the desired variables Also you will need to enter the Population Mean a value that you determine there is no variable designated for this in the Test Value field by clicking on the field and typing in the value iOptionsi the options button will present an options dialogue box where you can change your con dence limits something you don t really need to worry about and choose how you want to deal with missing data e g pairwise vs listwise see above test options for details 7Syntax to paste syntax to the syntax sheet leftclick the Paste button in the One Sample T Test box ito run the chisquare analysis without pasting to the syntax sheet leftclick OK Jndependent Sample T Test From the Analyze pulldown menu select Compare Means and then select Independent Sample T Test from the side menu This test requires you to identify your Continuous Dependent Variables in the Test Variables field and your DiscreteCategorical Independent Variable you can only test one grouping variable at time in the Grouping Variable field using the appropriate boxed arrows iDefming GroupsAfter Identifying the grouping variable you will need define your groups that is you have to tell the computer what the numbers are that you used to identify your two groups of interest Define your groups by leftclicking the Define Groups button In the Define Groups dialogue box you will have two options 1 Use specified values which you type into the grp l and grp 2 fields 2 Cut Point this option is for Independent variables that are continuous e g if you wanted 2 groups based on IQ one Below Average IQ and one Above Average IQ you could choose a cut point of 100 iOptionsi the options button will present an options dialogue box where you can change your confidence limits something you don t really need to worry about and choose how you want to deal with missing data eg pairwise vs listwise see above test options for details 7Syntax to paste syntax to the syntax sheet leftclick the Paste button in the One Sample T Test box ito run the chisquare analysis without pasting to the syntax sheet leftclick OK iPaired Samples T Test7 From the Analyze pulldown menu select compare means and then select Paired Samples T Test from the side menu This tests requires you to identify the time 1 and time 2 variables The first variable you select should be the pretest and the Measurement Concepts Chapter 5 PSYC 201 Psychological Research LEARNING OBJECTIVES De ne reliability of a measure of behavior a describe the difference between testretest internal consistency and interrater reliability Discuss ways to establish construct validity including predictive validity concurrent validity convergent validity and discriminant validity Describe the problem of reactivity ofa measure of behavior and discuss ways to minimize reactivity Describe the properties ofthe four scales of measurement nominal ordinal interval and ratio Reliability of Measures measure ofbe am If you weighed yourselfnow and then at 7 the end ofclass and you weighed the same both times you would say the scale is reliable Reliability refers to the consistency or stability of a h or Reliability of Measures con t Anymeasure True score Measurement error True score real score on the variable Reliable scores have considerably less measurement error Same person gets the test once per week for a year Raiiabie measure Number 0 scores obtained Unreliable measure 100 Score an test Less error Reliability of Measures con t I How do we assess reliability7 Correlation coef cients called regression coef cients Pearson ProductMoment Correlation coef cient o Symbolized as r A correlation assesses the relationship between variables Reliability of Measures con t I Acorrelation coef cient is used to compute the strength and direction of the relationship I Coef cients range from 000 to 100 and 000 to 100 Sign of the coef cient indicates direction Value ofthe coef cient indicates the strength 039 to 0 to 039 weak 079 to 040 and 040 to 079 moderate 100 to 080 and 080 to 100 strong Correlation coefficients Variables covary In opposite directions 100 000 100 Negative Variables covary In gt 4 lh gt e same direction N0 Positive relationship Scatter Diagram or Scatterplot for a positive correlation Positive Correlation As the measure for one variable increases the measure forthe other one increases eg as Q increases the Grade Point Average increases Positive Correlation an an inn 11m 12m 13a 14a 15m 15m LQ Correlation cont d r2 is the Coefficient of Determination The proportion ofthe total variability on that is accounted for by X Example for a correlation between IQ score and GPA in college If r 56 then r2 5656 31 This means that 31 of the variability in Y GPA in college can be accounted for by IQ scores The other 69 must be accounted for by other variables like motivation or testtaking ability etc Scatter Diagram for a negative correlation Negative correlation Negative Correlation As the measure for one variable increases the other variable decreases and gun eg As the number of hours of TV watched per day increases sixth grade reading skill decreases a 1 2 a a 5 s AmnunQ nfTV humd Reliability of Measures con39t To assess the reliability of a measure you need at least two scores on the measure from many individuals Methods of assessing reliability 1 Testretest reliability 2 Internal consistency reliability 3 lnterrater reliability Reliability of Measures con39t Re39iabi39ity 0f Measures com Test Retest Reliability Internal consistency reliability 39 Assessed by measuring the same individuals 39 Assess at one POW in time two points in time 39 All items should yield consistent results 39 Re39iabi39itymf ciem Shw39d be 80 hitherto be 1 Splithalf reliability correlation of individuals considered reliable total score on one half of the test with the total May want to use parallel forms score on the other half of the test 2 Cronbach s alpha correlation of each item with every other item then take the average Reliability of Measures con39t Increasing Reliability The more questions about a construct the higherthe lnterrater Reliability reliability 39 Correlation between the observations of raters The more variability on the construct Within the 39 A reliable measure mUSi Showa high sample of subjects the betterthe reliability agreement between raters orjudges Controlling the test situation such that the instructions are very clear and there are no distractions increases reliability Validity A valid survey or questionnaire measures what it is supposed to measure Content Validity Items are representative Questions are sensitive Is the measure that is used actually measuring Construct Validity of Measures I Construct validity refers to the adequacy of the operational defnition of variables the construct it is intended to measure Construct Validity of Measures con t I Indicators of construct validity Face many Ylw content or the measure arr5K5 m veiled we mm new measured cruenmannnmu r Scams an inn quotmagma am mm m a cnmncn an Malcolm or me construcll TYPE nlcntenon oriented many 1 cwevall iy rLsnmhnmeasuvcyrcdrcltmmw maer uvmnl validity mm m gmugs mm in mm an inn cansquot a score mmer an the inmasmc Convergent valldlly Snares m the inaasnm are terms to mixe measures at Hm same consumer Discriminant many Scores an lhw measure at nnlmlale to may measures mm is lncorclicatlv mum Construct Validity con t Convergent Discriminant or Concurrent Correlate the The questionnaire39s questionnaire39s results results should not with another correlate well With 3 established measure me of ofthe same construct or dimensi construct or dimension RESEARCH ON PERSONALITY AND INDIVIDUAL DIFFERENCES Systematic and detailed research on validity is most often carried out on measures of personality and individual differences Should use measures of personality that have demonstrable validity and reliability Example Mental Measurement Yearbook Reactivity of Measures Reactivity is a potential problem A measure is reactive if awareness of being measured changes an individuals behavior Falls to provide a measurement of the behavior under natural circumstances Reactivity of Measures con39t Use nonreactive or unobtrusive measures Minimize reactivity by Allowing time for individuals to become used to the presence of an observer or the recording equipment Obscuring the reason for a survey what the behavior of interest is or what relationship is being examined Carpet or tile wear 7 Smudges on glass 7 Archival data Variables and Measurement Scales Nominal Scales No numerical or quantitative properties Classlfles the levels of a variable into categoriesgroups Independent variables are often nominal or a categorical variable Variables and Measurement Scales con39t Variables and Measurement Scales con39t Ordinal Scales Rank orders the levels of the variable Allows categories to be ordered first to last highest to lowest biggest to smallest etc Quantitative but no values attached to the intenals Interval Scales Difference between the numbers is meaningful Intenals are equal in size 39 Quantitative but no meaningful zero reference point Variables and Measurement Scales con39t Identify the measurement scale for each of the following data Ratio Scales 39 Quantitative with all numerical properties including an absolute zero reference point PN 01 0 Circle your marital status Married Single Divorced Engaged Do you go to work Yes No If you work how many hours a week do you work Rate the extent you enjoy college on the scale below 1 2 3 4 5 Not Ver Much Very Much What is your class standing Freshman Sophomore Junior Senior What score did you receive on the verbal portion ofthe SAT Self Report Measures Measuring SelfReport Variables A The Survey Research Method Many participants are randomly eg stratified random sampling or otherwise nonprobablility sampling p140146 Provide Self reports of Behaviors Feelings current emotional states Attitudes andor Personality Constructs 1 Methodological Validity lnternal Validity Low this is correlational data External Validity Generalizability to Population High potentially Large Random Samples Generalizability to other Settings Mundane realism Low Experimental realism Low Pyschological realism should be high 2 Strengths More generalizable to the population as whole external validity though difficult and expensive to collect truly random samples often use stratified samples randomly sample subgroups of the population clustersampling Id clusters and randomly draw clusters 3 Problems with Survey Research a Information is less detailed than Obsenational and Case Study methods b It is oorrelational research Low lnternal Validity causeeffect relationships can not be established 0 Response styleset Response Acquiescence saying yes Response Deviation saying no Social Desirability not wanting to look bad or be uncooperative Crown Marlow Scale Self Deceptive Positivity Impression Management d Volunteer Problems people who want to participate agree to participate in research may be different from people who refuse more intelligent better educated more cooperative better adjusted desire more social approval B Psychometric Scaling amp SelfReport Methods Psychometric Scaling Definition developed between 1915ish 1930 s measure mental phenomenon with no clearly identifiable external cause but the experience can be reported by the individual eg feelings beliefs Intelligence and other mental processes 2 Summative Scales Measures of attitudes andor personality dimensions based on ratings of a series of statements selected to represent a specific topic Typically these scales use Numerical Rating Scales on a 5 or 7 pt scales Agree me nt 1 2 3 4 5 Strongly Disagree Disagree 1he Agree Strongly Agree Endorsement 1 2 3 4 5 6 7 Very Much Unlike Me Uhiike Me Uhiike Me Somewhat Neither Like Me her Not Like Me Somewhat Like Me Very Much Like Me Me Me The response numbers from all the items comprising a scale are summed and averaged Example Bloom County No Freaks Measure of Freakiness 1 I like limber Eskimos 2 It is okfor a person to have one or many tattoos 3 I would associate with people who gargle Windex 4 I would consider purchasing a Barm Manilow inflatable doll 5 It is acceptable to sleep with large tomatoes 6 feel that snake wrestling in JellO pudding is an acceptable social activity Ideally halfthe items are worded in the af rmative agreeing indicates more positive attitudes toward freaks halfthe items are worded in the negative disagreeing indicates more positive attitudes toward freaks Reverse Worded items must be reverse scored Balanced Example Reverse scored items 8 score ls become 78 28 become 68 33 become 48 etc The response numbers from all the items comprising a scale are summed and average Example Bloom County No Freaks Measure of Freakiness Score Reverse Score 1 I like limber Eskimos 2 It is inappropriate for a person to have one or many tattoos 3 I would associate with people who gargle Windex 4 I would not consider purchasing a Barry Manilow inflatable doll 5 It is acceptable to sleep with large tomatoes 6 feel that snake wrestling in JellO pudding is an unacceptable social activity C Measurement Reliability The consistency of a measure 1 Consistency Across Time TestRetest Give test at two points in time High positive correlations indicates good TestRetest Reliability usually about 50 over 3 months is average Not all psychological phenomena are expected to be stable Traits eg personality variables States eg transient moods Problem of Practice Effects Testing Effects Increase in performance due to item familiarity abilities knowledge and performance tests Solution Parallel Forms 2 Consistency Across Items Internal Consistency Cronbach s Alpha correlate all items in a scale with each other item intercorrelations All items should be responded to consistently highly correlated Adequate Consistency alpha or 3 70 Don t confuse Cronbach d with Type I error or D Measurement Validity Does the measure assess what it should A measure must be reliable in order for it to be valid reliability is a necessary but not suf cient criteria for validity 1 Content Validity Face Validity Does a measure contain the information it is supposed to a Representativeness are all the relevant domainsdimensionsconstructs represented Have experts and lay persons develop list eyeball it b Relevance are irrelevant domains dimensions constructs excluded experts eyeball it Ethics In Psychological Research Background Information Nazi Medical War Crimes Nuremberg Code Participation must be voluntary with informed Risk should be proportional to bene ts Freedom to withdraw Protect participants 39om harm Background Information Tuskegee Syphilis Study 19321972 National Research Act of 1974 National Commission forthe Protection ofHuman Subjects of Biomedical and Behavioral Research Belmont Report Respect for Persons Bene cence Justice The Common Rule 1981 Legal guidelines for federally funded research Background Information Human Radiation Experiments 19441974 Jewish Chronic Disease Hospital Study 1963 Willowbrook Study 19631966 APA Guidelines Ethical acceptability must be considered Primary concern should be minimal risk All involved must follow ethical guidelines Full disclosure of greater than minimal risk WN 01 Minimal concealment or deception with sufficient explanation ASAP APA Guidelines 7 Subject can withdraw at any time Protect from physical and mental discomfort and a Provide participant with true nature of study Remove adverse consequences and longterm effects 10 Keep participant confidentiality I 0 DHHS Guidelines Apply to all research involving human subjects with exception of research involving Standard educational tests Anonymous surveys Observation of public behavior Collection of study of pre existing data Institutional Review Board IRB Institutional Review Board Review scientific research involving humans in orderto protect the rights ofthe participants Ensures subjects have sufficient information to provide informed consent IRB must have A scientist nonscientist and someone not affiliated with the institution Both men and women What is At Riskquot Riskbenefit ratio Whether the benefits outweigh the risks to participants No Risk does not exist Is risk beyond what would be experienced in normal daily activities Types of Risk Physical Psychological Social harm Economic Legal Evaluation of Risk Evaluate risk in terms of Likelihood of occurrence Severity Duration after the research Reversibility Early detection 9 9 9quot Informed Consent DHHS guidelines require Statement that study involves research including a description ofduration and proce ures Statement of reasonable risks or discomforts Alternative procedures or treatments Bene ts of research Statement ofcon dentiality Statement of compensation or medical treatments should any injury occur Who to contact about participants rights Statement that research is voluntary and you can withdraw at any ime Deception Can only be used when benefits outweigh risk of deception Alternatives are not feasible Types of deception Active Passive Active Deception Misrepresenting purposes of research False statements of identity of researcher False promises made Violating promise to maintain anonymity Using confederates Using placebos Misleading settings for investigations Passive Deception Unrecognized conditioning Concealed observation Unrecognized participant observation Projective techniques or personality measures Comparing Means A Quick Guide to Z and ttests I The logic of Hypothesis Testing Revisited William Gosset the Guinness guy demonstrated that drawing samples from populations results in error small samples are more likely to demonstrate large sampling errors larger sample are less likely to demonstrate large sampling errors To determine if the pattern of data in our sample is meaningful likely to be found in other samples or if it is just the result of sampling error we use Inferential Statistical Tests of Hypotheses ie the Null Hypothesis HO no signi cant pattern and Alternative Hypothesis HA pattern is signi cant Statistical Tests quantify the pattern of data and then we compare that score to scores likely to have occurred by chance alone If we can be 95 or more con dent 5 or less uncon dent p g 05 that the pattern did not occur by chance alone then we say the pattern is Signi cant If we are less than 95 con dent more than 5 uncon dent p gt 05 that the pattern did not occur by chance alone then we say the patterns is Not Signi cant The type of Statistic you use depends on the type of data you have discrete vs continuous 11 Descrete IVs and Continuous DVs IE Scores for Groups Ztests and ttests allow us to compare the scores of groups to determine if they signi cantly differ Four basic hypotheses can be tested 1 Scores of a single sample group differ from the Population 2 Scores of two sample groups differ from one another 3 Scores from a single sample group on two separate occasions differ 4 Scores for Three or more sample groups differ from one another A Scores of a single Sample group differ from the Population Ztest compares a sample mean 7 to a population mean p when the population standard deviation 0 is known For Example we could compare the number of Cheezy Poofs eaten by a sample of RU students with average number eaten by the US Population Z Z sq lq Ho gt7 p HA 7 Critical Values come from the ZDistribution 7 055 chtical twotailed 196 I7 013 chticaltwotailed 258 If Zobtained is greater than or equal to Zmcal p g 05 then Sample Mean signi cantly differs from the Population Mean Reject Ho and Fail to Reject HA Single Sample t test if we don t know the population standard deviation 0 we can use the sample standard deviation s However we now use the t distribution the sampling distribution thank you William Gosset to test our hypotheses X 1 tsingleisarrgple s J Ho gt7 p HA 7 Critical Values come from the tDistribution with df nil we sampled one group so we loose 1 degree of freedom If tobmmd is greater than or equal to t ideal p E 05 then Sample Mean signi cantly differs from the Population Mean Reject Ho and Fail to Reject HA 1 Measures of Central Tendency Allow us to summarize an entire data set with a single value the midpoint 1 Mode The value score that occurs most often in a data set MoX Sample mode Mo Population mode 2 Median the point score which divides the data set in 12 e g 12 of the subjects are above the median and 12 are below the median Mdnx Sample Median Mdn Population Median 5 Mean the arithmetic average Directly considers every score in a distribution 2X Y Sample Mean X Ll 27 Population Mean n Z X 11 Skewed Distributions amp the 3M s Skewness refers to the shape of the distribution which can be in uenced by extreme scores Skewness is also an estimate of the deviation of the Mean Median and Mode Figure l Symmetrical Distribution Symmetrical Dist Mean Median Mode are all in the same location in the dist See Figure l Skewed Right Positively Skewed Mode in peak of distleft of center Median in center of distribution Mean in right tail of distribution See Figure 2 Skewed Left Negatively Skewed Mode in peak of dist 4ch right of center Median in center of distribution Mean in Mun left tail of distribution See Figure 2 Figure 2 Positively and Negatively Skewed Distributions Skcwcd right positive Skewed left negutivc 39t H Mo Mdn Mean Mean Mdn Mn I Measures of Variability Dispersion Like measures of central tendency measures of variability allow us to summarize our data set with a single value When central tendency and variability are considered together we get a more accurate picture of our data set The 3 main measures of variability Range Variance and Standard Deviation these formulas are the root formulas for many of the statistical tests that will be covered later eg ttests ANOVA and Correlation These measures tell us how much observations in a data set vary differ from one another that is how are they dispersed within the distribution Although this information is indirectly contained within measures of central tendency they don39t tell us much about the variance within our data Non Sequltur By Wlley OFFHN W VV 9M ONE 0 09 L K WW7 WE 39 MoTllF KOF NA umucwme 3 39 w 3 39 11 39 mmmmmgwwav Example I Number of miles traveled before traveling companion appears human n8 Zoo Penguins South Pole Penguins 5 2 5 3 5 4 5 5 5 5 5 6 5 7 5 8 EX 40 EX 40 Mean 5 Mode 5 Median 5 for both data sets They do not differ all zoo penguins hallucinate after traveling 5 mile while there is much more variability in the distances traveled by South Pole Penguins In order to draw accurate conclusions about our data both central tendency and variability must be considered 11 Range The numerical distance between the largest X maximum and smallest values X minimum tells us something about the variation in scores we have in our data or it tells us the width of our data set Range X maximum X minimum Range for Zoo penguins 55 O III Range for South Pole P39s 82 6 Problems with Range this is a summary measure that does not directly consider every value in the data set here only the two extreme numbers largest and smallest Therefore we do not know whether most of the scores occur at the extremes of the distribution or toward the center It is a very crude measure of variability For example Zoo Penguins South Pole Penguins North Pole Penguins 5 2 2 5 3 5 4 5 5 5 5 5 5 5 5 6 5 5 7 5 5 8 8 EX40 EX40 EX40 Range 0 Range 6 Range 6 Variance indicates the total amount of variability differences between scores in a data set by directly considering every observation To do this requires a point from which each observation can be compared to assess the amount they differ The Mean can be used as a point of comparison since it considers every observation in its calculation Mean Deviation Z X gt lt Zoo X Mean South Pole X Mean North Pole X Mean Penguins Penguins Penguins 5 0 2 3 2 3 5 0 3 2 5 0 5 0 4 1 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 6 1 5 0 5 0 7 2 5 0 5 0 8 3 8 3 2X 40 2X Mean 0 2X 40 2X Mean 0 2X 40 2XMean 0 Mean 5 Mean 5 Mean 5 Range 0 Range 6 Range 6 The sum of the mean deviation for any data set is always 0 This limits the usefulness of the mean deviation for summarizing different data sets with a single point If we square each deviation value then the negative values cancel out and we are left with a more meaningful value 0 X Mean X Mean2 South Pole X Mean X Mean2 Penguins Penguins 5 0 0 2 3 9 5 0 0 3 2 4 5 0 0 4 1 1 5 0 0 5 0 0 5 0 0 5 0 0 5 0 0 6 1 1 5 0 0 7 2 4 5 0 0 8 3 9 2X 40 2X Mean 0 2XMean2 0 2X 40 2XMean 0 2XMean2 28 Mean 5 Mean 5 Range 0 Range 6 North Pole X Mean X Mean2 Penguins 2 3 9 5 0 0 5 0 0 5 0 0 5 0 0 5 0 0 5 0 0 8 3 9 2X 40 2XMean 0 2XMean2 18 Mean 5 Range 6 If we sum these values we no longer get 0 but a number that re ects the total variance for this data set if we divide that number by N or n we get the average variance for this data set De nitional Population Formula O 2 Z X gt lt 2 N De nitional Sample Formula s2 Z X gt lt 2 nil Note sample variance uses nl rather than N because it is an estimate of the population variance Due to this reduced denominator the sample variance will always be slightly larger than the population variance 62 0 N s2 2X YY 2 2XY2 9 8 Zoo Penguins O O 2 2 20 n l 8 1 7 a Z N 235 South Pole Penguins 2 2ZX X g 4 s n l 1 7 X XZ 0222N 225 North Pole Penguins 2 X X 22 1 2E2257l4 n l 8 l 7 gives a good idea of how we get variance but it is time consuming for large data sets so we have developed mathematically identical algebraically equivalent formulas that are a little easier to calculate Computational Formulas Population Variance O 2 2X2 EX 2N N Sample Variance s2 2X2 EX Zn nl Note sample variance uses nl rather than N because it is an estimate of the population variance Due to this reduced denominator the sample variance will always be slightly larger than the population variance Zoo X2 South Pole X2 North Pole X2 Penguins Penguins Penguins 5 25 2 4 2 4 5 25 3 9 5 25 5 25 4 16 5 25 5 25 5 25 5 25 5 25 5 25 5 25 5 25 6 36 5 25 5 25 7 49 5 25 5 25 8 64 8 64 2x 40 40 40 2X2 200 228 218 EXHQX zooi 20mm 2 N 8 8 200 P 200 0 039 N 8 8 8 g 0 Zoo Penguins 2 X 2 2 Z 40 1600 2 2X 7 n 7 2007 8 7 2007 8 7 2007 200 20 nil 871 7 7 777 Z Z X2 E 1600 8 Z X 7 228 7 2287 Z N g 2287 200 28 7 35 U N 8 8 8 8 South Pole Penguins Z X Z z Z 40 1600 S2ZX7 n 72287 8 72287 8 zzgizooi n71 87 1 7 7 7 Z X2 L X 218 Loy 218 71600 U22 7 N 7 g 7 g 21872007E225 N 8 8 8 8 39 North Pole Penguins Z X 40 Z 1600 ZXZ GH 218 g 218 3 2187200 18 s2 25714 n7 1 87 1 7 7 7 Problems This formula is the base for many other statistical formulas however as a single summary measure it has little numerical meaning until it is converted to a standardized score Right now it represents the average distance each penguin is from the mean in squared mile units 3 Standard Deviation The square root of a variance The standardized variance value It provides us with a numerically meaningful measure of variance The average distance each observation is from the mean This value when combined with other stats methods allow us to infer what percentage of our observations are a certain distance from the mean Standard Deviation based on computational formula of variance Population St Dev Sample St Dev 2 S S The larger the value of variance or standard deviation relative to the numerical values of the observations the greater the amount of variability that is present in the data set Measures of Central Tendency and Variability Summarizing your Data for Others l Measures of Central Tendency Allow us to summarize an entire data set with a single value the midpoint 1 Mode The value score that occurs most often in a data set 0 ample mode Mo Population mode 2 Median the point score which divides the data set in A eg A of the subjects are above the median and A are below the median Man Sample Median Mdn Population Median 3 Mean the arithmetic average Directly considers every score in a distribution V gt T 7 h A Sample Mean 71 H Populauon Mean 7 21X X X n N 7 Skewed Distributions amp the 3M s Skewness refers to the shape of the distribution which can be influenced by extreme scores Skewness is also an estimate of the deviation of the Mean Median and Mode Symmetrical Dist Mean Median Mode are all in the same location in the dist Skewed Right Positively Skewed Mode in peak of disteft of center Median in center of distribution Mean in right tail of distribution skewed right mum Mo Mdn Mean Skewed Left Negatively Skewed Mode in peak of dist right of center Median in center of distribution Mean in left tail of distribution Slmwd Ic ncgaiiw Mean Mdn M 0 l Measures of Variability Dispersion Allow us to summarize our data set with a single value Central Tendency Variability a more accurate picture of our data set The 3 main measures of variability Range Variance and Standard Deviation These formulas are the root formulas for many of the statistical tests that will be covered later t test ANOVA and Correlation Tell us how much observations in a data set vary differ from one another How are they dispersed within the distribution Non Sequitur OFFLWV 39v 9N ONE o L Although measures of central tendency tell summarize some aspects of our data they don39t tell us much about the variance within our data Example Number of miles traveled before traveling companion appears human n8 Mean 5 Mode 5 Median 5 for both data sets They do not differ all zoo penguins hallucinate after traveling 5 mile while there is much more variability in the distances traveled by South Pole Penguins In order to draw accurate conclusions about our data both central tendency and variability must be considered Zoo Penguins South Pole Penguins 5 2 5 3 5 4 5 5 5 5 5 6 5 7 5 8 XX 40 XX 40 Range The numerical distance between the largest X maximum and smallest values X minimum tells us something about the variation in scores we have in our data or it tells us the width of our data set Range X maximum X minimum Range for Zoo penguins 55 O Range for South Pole P39s 82 6 Problems with Range Does not directly consider every value in the data set here only the two extreme numbers largest and smallest We do not know whether most ofthe scores occur at the extremes of the distribution or toward the center For example Zoo Pengmus i 5 m u m u JI u EX 40 Range 0 South Pole Penguins u 4 Lu Ix chm 8 EX 40 Range 6 North Pole Peugums 00 u u u u u u J EX 40 Range 6 lll Variance indicates the total amount of variability differences between scores in a data set by directly considering every observation Requires a point from which each observation can be compared to assess the amount they differ The Mean can be used as a point of comparison since it considers every observation in its calculation Mean Devianon XX X Y Zoo X Mean South Pole X Mean North Pole X Mean Penguins enguins Penguins 5 0 2 3 2 3 5 0 3 2 5 O 5 0 4 1 5 O 5 0 5 0 5 O 5 0 5 O 5 O 5 0 6 1 5 O 5 O 7 2 5 0 5 0 E 3 8 3 2x 40 X Mean o x 40 x Mean o x 40 x Mean 0 Mean 5 Mean 5 ean 5 Range 0 Range 6 Range 6 The sum of the mean deviation for any data set is always 0 This limits the usefulness of the mean deviation for summarizing different data sets with a single point if we square each deviation value then the negative values cancel out and we are left with a more meaningful value o X Mean X Mean2 South Pole X Mean X Mean2 Penguins Penguins 5 O O 2 3 9 5 O O 3 2 4 5 O 0 4 1 1 5 O 0 5 0 0 5 Cl 0 5 0 0 5 Cl 0 6 l 1 5 O 0 7 2 4 5 0 Cl 8 3 9 EX 40 2XMean 0 2X Mean2 D EX 40 2XMean 0 2XMean2 28 Mean 5 Mea 5 Range 0 Range 6 North Pole X Mean X MeanZ Penguins 2 3 Q 5 O 0 5 O 0 5 0 O 5 O 0 5 0 0 5 0 0 B 3 9 EX 40 ZXvMean 0 ElXMean2 18 Mean 5 Range 6 lfwe sum these values we no longer get 0 but a number that reflects the total variance for this data set if we divide that number by N or n we get the average variance for this data set Definitional Population Formula 02 2X Mean2 N Definitional Sample Formula s2 ng Mean2 n 1 Note sample variance uses n1 rather than N because it is an estimate of the population variance Due to the smaller denominator the sample variance will always be slightly larger than the population variance Zoo X Mean X Mean2 Penguins Ammmmmmmm 0000000 0000000 M X I D 0 W Mean 5 Range 0 Zoo Penguins South Pole X Mean X Mean2 ngums 2 3 9 3 2 4 4 1 1 5 0 O 5 0 D 6 T 1 7 2 4 8 3 9 2X 40 2XMean 0 ZgtltMean2 28 Mean 5 Range 6 South Pole Penguins North Pole X Mean X Mean2 Penguins 2 3 9 5 U 0 5 0 0 5 0 0 5 O 0 5 O 0 5 D 0 8 3 9 2X 40 2XMean 0 Zgtlt Mean2 Mean 5 Range 6 Nonh Pole Penguins 18 Definitional Formula is time consuming for large data sets We have developed mathematically identical algebraically equivalent formulas that are easier to calculate Computational Formulas Population Variance 02 2X2 ZX 2N N s2 2X2 ZX 2n n1 Note sample variance uses n1 ratherthan N because it is an estimate ofthe population variance Due to this reduced denominator the sample variance will always be slightly largerthan the population variance Sample Variance Zoo Penguins South Pole Penguins North Pole X2 Penguins Zoo Penguim Zoo lt2 South Pole lt2 North Pole X2 Penguins Penguins Penguins 5 25 2 4 2 4 5 25 3 9 5 25 5 25 4 16 5 25 5 25 5 25 5 25 5 25 5 25 5 25 5 25 8 3G 5 25 5 25 7 49 5 25 5 25 8 64 8 64 XX 40 4O 40 2X2 200 218 A amp m 1600 7 3 n M 2287 200 28 South Pole Penguins 87 40V 1600 7 s Zoo lt2 South Pole lt2 North Pole X2 Penguins Penguins Penguins 5 25 2 4 2 4 5 25 3 9 5 25 5 25 4 16 5 25 5 25 5 25 5 25 5 25 5 25 5 25 5 25 6 36 5 25 5 25 7 49 5 25 5 25 8 64 B 64 2X 40 4D 40 XX2 200 228 218 J Y 401 160 Z 218 T 218 T 2187200 18 C 7 iiii 225 N 8 8 8 S Nonh Pole Penguins 21 Problems This formula is the base for many other statistical formulas however as a single summary measure it has itt e numerical meaning until it is converted to a standardized score Right now it represents the average distance each penguin is from the mean in squared mile units Standard Deviation The square root of a variance The standardized variance value It provides us with a numerically meaningful measure of variance The average distance each observation is from the mean This value when combined with other stats methods allow us to infer what percentage of our observations are a certain distance from the mean 9quot Standard Deviation based on computational formula of variance Population St Dev Sample SI Dev l IEXl 7 7quot ZX1 7 if 0quot V s A39 n 7 1 a 02 S Zoo Penguins T V th respect to sample standard deviations s we can say Zoo penguins are an average ofO miles 39om the mean number of miles walked before hallucinating SZXS7 2 V th respect to sample standard deviations s we can sa South Pole penguins are an average of2 miles from the mean number of miles walked before hallucinating Measures of Central Tendency and Variability STAT Chaps 4 amp 5 PSYC 201 Psychological Research Measures of Central Tendency Summarizing an entire set of numbers with one single score that is located in or near the center of the distribution Mode Median Mean Mode The mode is the most frequently occurring score in a distribution This is easy to find once you have sorted the data from lowest to highest scor There may be more than one mode for a distribution of data Median The median is the value in a distribution that has 50 ofthe cores falling above it and 50 ofthe Scores falling below it To calculate the medial l first ou put the scores h orderfrom lowest to highest or highest to lowest rank orderthern or sort t erh thto asce dll ig or deSCel idll ig order The median is the Score that falls atthe middle point in the distribution To nd the medial l of al l eveh numberofscores Wit ho duplicate scoresi the middle first put the scores ll l ordertrorh low to high The value halfway between the middle two scores is the medial l add the middle two scores and divide by Median cont d To calculate it when the scores are in a 39equency distribution Score n Mdn LRL 50N SFB x h Where LRL Lower real limit ofcritical interval N total number ofcases SFB sum of 39equencies below critical interval f frequency wi hin critical interval h Interval siz Mean The mean is the balance point in a distribution39 It is the most average score in a data set the arithmetic average sample mean M a The sum ofthe scores divided by the number of scores The raw average is Weighted Mean The m h 39 39 39 account the number of credit hours each grade was worth or group size ifthis were a frequency distribution weighted mean MW M SW 80 what is the weighted mean for the student39s grades If the distribution is skewed which is the better measure of central tendency an M 4 mm hnuinghtt wmnr Amman mt Measures of Variability a y z scores in a distrlbutlon dlffer or vary from each other howthey are dispersed Range Mean Deviation aka Average Deviation Variance Standard Devtatton Range Tne distance between tne argest and smaHest ubsewatmns m a data set Range R xwexmm Advantages Easytu umpute Easytu understand Disadvantages msensmve tn tne msmbuuun ur scares wmnn tne Wu extreme e 9 22mm vs 11 A A n 1 7 new have R e Mean Deviation 0r Average Deviation Measures of variability are supposed to provide information about how good a descriptor the average or mean is ofthe data set So why not use the average deviation of scores from the mean as a measure ofvariability ADZ X M N The problem is that the sum of each score minus the mean equals zero always Variance To avoid the zero problem we square each of the deviation values then sum them up and divide by the numbero s SZZXM2 cr2ZXM2 N1 N Computational formula for sample variance 32 20 Ex 2 N N1 Variance cont d Advantages Uses all the data Very useful in inferential statistics because of its mathematical properties ie the Principle of Least Squares Disadvantages Not so good as a descriptive statistic because the value is not intuitive it is expressed in squared units Standard Deviation The standard deviation is a standardized score which is the average deviation about the mean It is the square root of the variance Sample standard deviation s the square root 0 52 Population standard deviation o the square root 0 0 2 Standard Deviation cont d Advantages Uses all the data It is widely used Very useful in inferential statistics Not expressed in squared units so makes more sense descriptively Disadvantages Example problem tat ngl ll ii 0 20 20 tnat her students often 1 21 21 Complall l dunng class about 3 21 24 ll lvolvll lg penguins sne decides to assess tne 4 22 7 frequency of penguin 22 9 nallucinations for students 23 9 for students dunng 25 1 introductory Englisn class 26 4 27 5 29 9 20 30 20 39 The Use of Physical Ability and Fitness Testing for Law Enforcement Occupations Michael G Aamodt PhD Department of Psychology Radford University Radford VA 241426946 m aam odtradford edu Paper presented at the annual meeting of the Society for Police and Criminal Psychology October 1994 Madison Wisconsin Considerations for Physical Ability and Fitness Requirements 0 Job relatedness of the actual requirement Essential Linearly valid Preferred Neither necessary nor useful 0 Level of skill or ability Content validity Use of norms Scale A Importance and Skill Level Lquot Officer must be able to perform this task at a high level to ensure the safety of the public or the officer 4 Performance of this task at a high level is necessary to meet the minimum requirements of the job L Performance of this task at a high level though not required will result in an officer receiving a higher performance appraisa Performance of this task at a minimal level is required to ensure safety or the proper performance of the job i Performance of this task would be useful but is not essential not will it result in higher performance ratings O This task is not performed in our department Considerations for Physical Ability and Fitness Requirements The time at which the skill must be present Time of testing Time of hire Start of the academy Start of field training After promotion or reassignment Never Scale B When Skill Must be Present 4 At the time of hire LA At entry into the academy N Upon academy completion and before the start of eld training After promotion or reassignment 0 Never Considerations for Physical Ability and Fitness Requirements 0 Availability of alternatives resulting in less adverse impact Job redesign Push bumpers on patrol cars Increased use ofpepper spray Pretesting interventions Practice tests Physical tness programs Physical Ability Measurements Job simulation General ability measures e g situps pushups JobRelated Purpose Catching eeing suspects Selfdefense Firing a weapon Moving Victims to safety Changing tires Physical Fitness or Wellness 0 Measurements Cardiovascularaerobic fitness Body fat norms Flexibility norms JobRelated Purpose Officer safety following a pursuit Decreased health costs Increased public image Research in Psychology I Research Proposal Guide Part II Title How to Write a Research Paper Methods Results Discussion and Conclusions Without Developing an Overwhelming and Compulsive Urge to Kill Author Jeff Aspelmeier Institution Radford University formerly the Radford Institute for the Criminally Inane I Methods A Participants Report information regarding subjects here 1 number of subjects 2 demographic characteristics 7 gender number or of each 7 ethnicity African American Asian Paci c Islander Native American Hispanic andor Latino Caucasian and other age range average age Class rank if college students are used relationship status if applicable single dating engaged married separated divorced 3 describe where from how selected how assigned to groups if applicable and incentives for participation eg payment or course credit For Example note that this should be double spaced Methods Participants Participants were 120 undergraduate college students attending a medium sized Southeastern university who were given course credit for their participation Participants ranged in age from 18 to 26 with a mean age of 1894 A majority of the participants were female 70 and 30 were male Also a majority of the participants was Caucasian 858 92 were AfricanAmerican 1 were AsianPacific Islander less than 1 were NativeAmerican less than 1 were Hispanic and 25 reported other ethnicity A majority of the participants were freshmen 858 108 were sophomores 17 were juniors and 17 were seniors The average GPA reported was 316 with a range of 230 A majority of the participants were single 925 33 were married 25 were divorced 17 were engaged 4 If they have important characteristics describe them eg depressed or ADHD and how determined 5 if participants excluded explain why and describe criteria for inclusion in the study Also report final sample size eg Two participants were excluded from the study do to their lack of hepatic tissue no liver and their advanced state of death The remaining sample consisted of 178 participants B Materials Measures Apparatus 1 If you are using paper pencil tests questionnaires each one used should be described in detail and include examples of items a description of how measures were computed from the questionnaires the mean the standard deviation and the range Also for scales with multiple items the Cronbach s Alpha should be reported For Example Note this should be double spaced Measures A measure of fearful animal attitudes was obtained using Aspelmeier s 2002 Radford Avoidant Beast Interaction Test RABIT which assesses the degree of participant s negative attitudes regarding small fury animals and their perceived likelihood of avoiding interactions with small fury animals Participants rated 12 items on a seven point numerical rating scale as to how descriptive they were of them 1 very undescriptive of me 7 very descriptive of me For items one through six ratings were scored and summed such that a higher score indicated more negative attitudes toward small fury animals NATSFA withM 455 SD 212 and range 699 Cronbach s Alpha an estimate of internal consistency was 89 Examples of the NATSFA scale items are 1 The Easter Bunny makes me sweat and 2 I often feel that vicious rabbits are lurking in the shadows For items seven through 12 ratings were scored and summed such that a higher score indicated a greater perceived likelihood that one would avoid interactions with small fury animals AISFA withM 389 SD 257 and range 685 Cronbachs Alpha was 88 Examples of the AISFA scale items are 1 I would probably never go to a park that did not implement squirrel control techniques and 2 I would never wear a baby seal fur coat for fear of being attacked by it 3 If you are using some kind of equipment or computer software to test participants then describe the equipment fully Note sometimes this can be embedded in the procedures especially if your IV depends on how the equipment is set up eg group 1 gets set up A and group 2 gets set up B For Example Apparatus A second measure of small fury animal phobia was obtained by using an armpit emissions assessment A standard 400 mhz PC was programed to present various photographs of inanimate objects and a variety of photographs and cartoon caricatures of small woodland creatures Participants arm pits were fitted with electronic moisture collection cups Model Number THX1138 Lafayette Instrument Co Lafayette IN These cups records the amount of sweat produced in each armpit in milliliters ml The sweat emissions from each pit were averaged It should be noted that it was requested that participants avoid use of antiperspirants for at least three days prior to testing Scores were taken before and after exposure to the stimulus and a difference score was calculated according to the formula Post Test Pre Test such that a higher score indicated that participants sweat production increased after exposure to the test stimulus The average amount of sweat production across all types of photographs was M 156 SD 1254 range 6485 A majority ofthe participants 62 had no increase in sweat production for any of the pictures C Procedure 1 Include a complete description of what happened to a typical subject in chronological order from beginning to end If appropriate include any unexpected additions to the study 2 Include the description of the design experimental quasiexperimental longitudinal etc 3 Provide an operational definition of the IV this definition should be the most descriptive one given in the paper 4 Provide an operational definition of the DV this def Should be the most descriptive one given in the paper Describe how changes in the DV will be observed and recorded For Example Procedures Participants initially agreed to spend two consecutive nights in the Radford Animal Avoidance ResearchCenter RAAR After receiving informed consent a catheter was surgically inserted into the participant s galbladder Over the first night of testing hepatic secretions were measured The average rate of bile production was recorded in milliliters per hour M 32 SD 18 range 135 After the first night the catheter was removed and participants were allowed to continue with their daily routine until 930 pm at which time they returned to the lab for further testing During the second night of testing pancreatic secretions were measured Participants blood sugar levels were measured every hour in order to establish each individual s rate of insulin production measured in micrograms per hour M 125 SD 72 range 50 After the second night of testing participants were given both the RABIT and the Pit Sweat measures of small fury animal phobia After completing the measures participants were thanked for their participation and asked if they had any questions or concerns It should be noted that during the second night of testing it was discovered that several participants 33 were not secreting insulin do to diabetes It was decided not to exclude these participants in that it would be useful to compare these participants with nondiabetic participants with respect to small fury animal phobia II Results note on your paper the results title will be centered and will not have a roman numeral beside it A This section contains all of the results but no conclusions 1 order Descriptive statistics first Tests with Demographic Variables second and Inferential statistics second B Descriptive Data Here we present the either the group frequencies for Discrete variables or means standard deviations and ranges for Continuous variables for all variables unless already provided in the Methods section as was done here C Demographic Analyses The purpose of these analyses is to establish that your demographic variables are not contributing to or confounding the associations we find between the Main Vars ie IV s and DV s For this paper the demographic analyses will be the first part of the results section a Tell the reader what variables were tested and which analyses were significant if any Example 1 If no Significant Associations were found Results Demographic Analyses In order to identify associations between demographic variables age GPA sex ethnicity and class rank and the main variables of interest bile production rate insulin production rate diabetics vs nondiabetics Pit Sweat volume change Pit sweat increase vs no pit sweat increase NATSFA and AISFA a series of preliminary analyses were conducted None of the preliminary analyses were significant The demographic variables were excluded from further analyses b When you have significant associations tell the reader what variables were associated and report the statistic Also explain to the reader what the statistics mean by referring to people and their behavior Example 2 If significant Associations were found Results Demographic Analyses In order to identify associations between demographic variables age GPA sex ethnicity and class rank and the main variables of interest bile production rate insulin production rate diabetics vs nondiabetics pit sweat volume change pit sweat increase vs no pit sweat increase NATSFA and AISFA a series of preliminary analyses were conducted A significant positive correlation was found between GPA and pit sweat volume change rl 18 56 p lt 001 Participants with higher grade point averages tend to show higher levels of armpit perspiration following exposer to photos of small woodland creatures Also participants who showed an increase in armpit perspiration following exposure to photos had significantly higher GPA s compared to participants who did not show an increase in armpit perspiration tll8 345 p 006 Means Standard Deviations for the pit sweat increase group and the no pit sweat increase group were 322 3235 and 288 3756 respectively Finally there was a significant association between sex and pit sweat increase vs no pit sweat increase X2l N l20 5185 p lt 001 Specifically females tended not to show a pit sweat increase after exposure to photographs while males were much more likely to show an increase in armpit sweating during the procedure See Table l for crosstabulations None of the remaining analyses were significant D Main Analyses Here we restate the hypotheses between the main variables describe your hypotheses with respect to the relationships between variables and scores tell what statistics were used to test this hypothesis and then give the results of the test and describe the behavior Example Main Analysis To test the hypothesis that hepatic secretions would be associated with self reports of small fury animal phobia a series of correlations between Bile Production and scores on the RABIT self report were computed Bile Production was signi cantly positively associated with the NATSFA subscale and the AISFA subscale r 176 49 p lt 001 r 176 67p lt 001 respectively Participants who produced greater amounts of bile reported more negative attitudes toward small fury animals and that they were more likely to avoid interactions with small fury animals Also It was hypothesized that higher scores on the self report NATSFA subscale of the RABIT would be associated with greater amounts of pit sweating but only when the visual stimulus depicted a small woodland creature To this end the pit sweating volume change of participants scoring above the mean on the NATSFA was compared with that of participants scoring below the mean 60 participants were included in each group across the 3 stimulus conditions Inanimate Objects Animal Pictures and Animal Caricatures To test this hypothesis Oneway Anova s were computed first for the total sample irrespective of NATSFA score and then separately for the participants scoring above and below the mean on the NATSFA For the total sample there was a significant effect for the stimulus condition F 2 174 667 p lt 001 Results of Fisher LSD posthoc tests revealed that photographs and caricatures of small animals elicited more pit sweating than inanimate objects with means and standard deviations of 2000 534 195 513 and 300 522 respectively Also participants with high NATSFA scores showed more pit sweating in the caricature and photo conditions than in the inanimate object condition See Table 2 Further participants scoring low on the NATSFA did not differ in pit sweating across the 3 conditions Figure 1 displays group means graphically 7 Note that the preceding paragraphs are both examples where the test statistics are reported in the text and examples where the test statistics are reported in a table and figure When you have several statistical tests that are very similar it is often preferable to put the data in a table You can do either but the stats must be reported somewhere Further even if you put the stats in a table you must describeexplain the results in the text Remember that your explanations should focus on people and their behaviors rather than variables and scores 7 Also for the more advanced statistical users you may have noticed that the hypothesis tested in paragraph two of the example above would really best be tested using a Twoway Anova Factoral Anova rather than a series of Oneway Anovas See the Appendix of this handout for an example of how to report the same results tested with a Twoway Anova Example Continued With respect to hepatic ecretinn it was 1r quot 39 A that bile r J quot would be associated with behavioral measures of animal attraction Bile production was significantly positively correlated with pit sweat volume change r 176 39 p lt 001 Also Participants who showed an increase in pit sweat volume when pictures of small woodland creatures were displayed produced significantly more bile than nonpit sweat increasing participants t 176 367 p lt 001 Means and standard deviations in parentheses for pit sweat and nopit sweat groups were 645 151 and 215 5995 respectively With respect to pancreatic secretions it was hypothesized that insulin production would be associated with self report measures of animal attraction Contrary to the expected results no significant associations were found between insulin production rate and the NATSFA or AISFA measures r118 14p 23 ns and r118 11p 58 713 respectively You will notice that not all the possible hypotheses are tested in this handout These are omitted to conserve space in the handout However your paper should report all the statistics for all of the hypotheses tested Regardless of whether they are significant or not 11 Discussion A This section contains the conclusions that can be drawn from the results of your data analysis 1 Start by once again restating the studies hypothesis It should be more general than the description you gave in the previous sections Talk about people and behaviors or subjects and behavior 2 Highlight the hypotheses that were supported 3 Suggest reasons as to why some hypotheses were not supported if relevant 4 Discuss the strengths and limitations of the study you report Focus on Measurement Validity Internal Validity and the Various components of External Validity Generalizability to the population Mundane Realism and Ecological Experimental Realism 5 Discuss how your results inform the psychological community with respect to the topic of research 6 Discuss suggestions for future research 7 Final statement needs to address how the present study affects our understanding of the universe andor the condition of humans in the universe Example the numbers in parentheses in the text below are for your bene t and should not be included in the text of your manuscript Discussion 1The present study tested the hypothesis that people who have greater hepatic and pancreatic secretion output would report more negative attitudes toward small fury animals and be more likely to demonstrate phobic responses to small fury animals 2Results support the hypothesis that hepatic secretions are associated with small fury animal phobia However little support was obtain for the 39 J I quot 39 that J quot quot are 39 A with fur related phobias 3 This unexpected result can be interpreted in several ways It may be that there truly is no link between insulin production and fear of small fury animals Altemately it may be that there is an association but the present study s design was not sensitive enough to identify the association due a variety of potential factors First these finding may re ect sample problems That is the present study s focus on a college population severely limits the generalizability of the results It may be that other more stratified samples would show the predicted insulin fur phobia link Also it has been noted that unique eating and drinking habits of college students can in uence measures of insulin production Budweiser Miller amp Daniels 1990 Second the present study s use of nocturnal pancreatic emissions may not have been appropriate It has been noted that metabolization of sugar is lowest during the sleeping hours Hershey amp Nestle 1952 Use of daytime pancreatic secretions would be need to adequately test this hypothesis Third neither the RABIT nor the Pit Sweat paradigm have been validated using other measures of small animal phobias While they appear to have face validity it may be that these measures only tap select aspects fur phobia This is important to the present study in that several researchers have noted that some animal phobics tend to show erratic and inconsistent phobic responses to the same stimulus Sylvester Granny amp Tweety 1967 Such periodicity in phobic behavior may re ect the periodicity of pancreatic secretions The design of the present study does not allow for the testing of this hypothesis 4 Though this study does suggest that hepatic secretions may be associated with animal phobia causal links can not be established An uncontrolled third variable may be confounding these results For example spleen size was not measured and controlled for in these analyses Further it may be that more psychological factors may be in uencing this processes studied here especially considering that several psychologists and biologists have commented on the connection between mind and body Pebody amp Sherman 1968 Flinstone amp Rubble 1962 Mephisto amp Kevin 1999 In conclusion 5 the present study is important in that it provides support that small fury animal phobia has its roots in organic tissues outside of the spleen an idea that was pure speculation prior to these findings 6 Future research should direct attention to both psychological and biological factors that in uence small fury animal phobia Also future research may want to test more experimental designs For example regulating hepatic and insulin output through the use of randomly assigned treatment conditions Such a line of research may make it possible to treat individuals who suffer from maladaptive levels of romantic animal phobia 7 This line of research is crucial to developing our understanding of the dynamics of small fury animal phobia and developing public and mental health policies aimed at protecting our citizens from cute woodland creatures witch posse numerous threats to our culture and ecosystem 7 Note that the references would be included next but are not presented in this guide Appendix Example Reporting the Results of a Twoway Anova Also It was hypothesized that higher self reports on the NATSFA subscale of the RABIT would be associated with greater amounts of pit sweating but only when the visual stimulus depicted a small woodland creature To this end the pit sweating volume change of participants scoring above the mean on the NATSFA was compared with that of participants scoring below the mean each group consisted of 60 participatns across the 3 stimulus conditions Inanimate Objects Animal Pictures and Animal Caricatures A 2 High vs Low NATSFA score x 3 Stimulus Condition mixed Anova design was computed There was a signi cant main effect for both NATSFA score F1 174 558 p lt 001 Participants with high NATSFA scores showed more pit sweating than low scoring participants with means and standard deviations of 2533 523 and 300 501 respectively Also there was a significant main effect for the stimulus condition F 2 174 667 p lt 001 Results of Fisher LSD posthoc tests revealed that photographs and caricatures of small animals elicited more pit sweating than inanimate objects with means and standard deviations of 2000 534 195 513 and 300 522 respectively Finally a significant interaction was found between the NATSFA score group and the stimulus type F2 174 2020p lt 001 Results oftests of simple effects revealed that participants with high NATSFA scores showed more pit sweating in the caricature and photo conditions than in the inanimate object condition See Table 3 Also participants scoring low on the NATSFA did not differ in pit sweating across the 3 conditions Further Participants exposed to either the caricature or the photos with High NATSFA scores demonstrated more pit sweating than participants reporting Low NATSFA score Finally participants exposed to the inanimate objects who reported high NATSFA did not differ from participants reporting low NATSFA Figure 1 displays group means graphically Table 1 Crosstabulatz39on of Sex and Pit Sweat Increase vs No Pit Sweat Increase Groups Pit Sweat Group Classi cation Increase N0 Increase X2 df V Males 2583 417 5185quot 1 65 236 236 1167 5833 336 336 Note W p lt 001 Standardized Expected Residuals appear in parentheses bellow means Table 2 Mean Pit Sweat Volume for Participants with High NATSFA Scores and Participants with Low NATSFA Scores Separate for Each Stimulus Condition Stimulus TVDe Caricature Picture Inanimate Object F High NATSFA 3700b 3600b 300a 1004 541 522 500 Low NATSFA 300a 300a 300a 001 538 499 501 Note df 2 57 for both analyses p g 01 Means within rows with differing subscripts are signi cantly different at least p g 05 with respect to Fisher s LSD post hoc analyses Table 3 Mean Pit Sweat Volume for Participants with High NATSFA Scores and Participants with Low NATSFA Scores Separate for Each Stimulus nnditinn example ofa TwoWay 39 quot Stimulus TVDe Caricature Picture Inanimate Object F High NATSFA 3700b 3600b 300a 1004 541 522 500 Low NATSFA 300a 300a 300a 001 538 499 501 F 2003 1802 003 Note dffor Stimulus Type simple effect 2 174 dffor NATSFA simple effects 1 174 p g 01 p g 001 Means within rows with differing subscripts are significantly different at leastp 05 with respect to Fisher s LSD post hoc analyses Post Hoc results for simple effects in columns are not displayed Chisquare Goodness of Fit Test The chisquare test is designed to test differences whether one frequency is different from another frequency The chisquare test is designed for use with data on a nominal scale when all you know about people is the category they39re in The chisquare test gives you a way of comparing the results that you should get ifthe number of passengers has no effect on the frequency of stopping to the results you actually got For example Having a coin come up heads or tails is a variable on a nominal scale Heads is a different category from tails If we ipped a coin 20 times how many times should it come up tails 10 Say the actual number of tails was 18 Is this number far enough away from 10 for you to be suspicious that the coin was rigged The chisquare test tests whether the odds are less than 5 of getting 18 tails when you know that 10 tails is what you would expect to get What we39re going to do is to calculate a value for the chisquare statistic and then compare it to a value for chisquare that we can look up in the back of the stats book That number will tell you how large the calculated value has to be in order to be con dent that the result was not just due to chance To calculate the value for chisquare for the coin data you need to know what frequencies you would expect to get 10 heads and 10 tails You know that the actual values were 2 heads and 18 tails To get chisquare 1 For every possible outcome subtract the expected frequency from the observed or actual frequency 2108 18108 2 Now square each of these differences 64 64 3 Divide each of these squared differences by the original expected frequency for that cell 6410 64 6410 64 4 Add these numbers up This gives you the value for chisquare 128 The equation for calculating Chisquare is thus 2 fa13902 X 2 re The steps for calculating in table for are f0 fe f0fe f0fe2 f0fe2fe Heads 2 1o 8 64 64 Tails 18 10 8 64 64 x2 128 To get the critical value for Chisquare 1 You need to know the number of degrees of freedom This is equal to number of possible categories minus one 21 1 2 You need to know which column to look the value up in lfyou are going to use 5 as the odds of the frequencies being different by chance look the number up in the 05 column 348 In this example the number for chisquare that we calculated was larger than the comparison number we looked up in the book This tells us that there is less a 5 chance that the percentages are different by chance 80 you might decide that the coin is rigged APA format for writing the conclusion statement is The frequencies of the tails option in the coin ips is significantly greater from what one would expect to get by chance X2 1 N 20 128 p lt 05 A Chisquare test when there is one variable is often referred to as a Goodness of Fit Test ChiSquare Test of Independence Now let s say we observe drivers at an intersection with a stop sign We record two pieces of information whether the driver stops at the stop sign and whether the driver is talking on a cell phone The question we want to answer is whether drivers are less likely to come to come to a complete stop at the stop sign when talking on a cell phone compared to when drivers are not talking on a cell phone Here are the data Cell phone No cell phone stop 5 15 2o dont stop 20 10 30 25 25 N50 The question is really whether the percentage of drivers who stopped while talking on a cell phone is signi cantly different from the percentage ofdrivers who stopped while not talking on a cell phone Obviously 525 or 20 of drivers with no passengers came to a complete stop 1525 or 60 of drivers with one or more passengers came to a complete stop The chisquare test will still tell us if the observed frequencies are signi cantly different from the expected frequencies We can use the numbers in the row and column margins to help us to compute the expected frequencies The expected frequency for each cell is computed using the following formula fe row totalcolumn total N For the top left cell the expected frequency is 2025 50 50050 10 top right 2025 50 50050 10 bottom left 3025 50 75050 15 bottom right 3025 50 75050 15 Once you have the expected frequencies the calculation for chisquare is exactly the same except that you have four rows in the calculations instead of two f0 fe f0 f62 f0 f 312 fe Stop no pass 5 10 5 25 25 Stop pass 5 10 5 25 25 No stop no pass 20 15 5 25 167 No Stop pass 10 15 5 25 167 834 You compare this observed value for chisquare against a comparison value you look up in the chisquare table You still use the 05 column because we39re using 5 as the odds we39re using to make our decision The number of degrees of freedom to use is found from the following equation df of rows 1 of columns 1 In this case we have 2121 11 1 giving us a comparison value of 384 We can say that the percentages found in the first column are signi cantly different from the percentages found in the second column Drivers on a cell phone are significantly less likely to stop at a stop sign than drivers not talking on a cell phone X2 1 N 50 834 p lt 05 ChiSquare Practice Problems For each of the problems below b a c d i Algebra l No algebra l Please state both the null and alternative hypotheses for this question Provide the decision rule for making this decision Use an alpha level of 05 Show all of the work necessary to calculate the appropriate statistic What conclusion are you allowed to draw Write a conclusions sentence in APA format ie l N E 4 5 t includes the appropriate statistical information From Thome 1989 At a state university the student population is approximately one third male and twothirds female Over a twoday period the gender of each student entering the student union is recorded with the following results males 452 females 1548 Determine whether males and females are as likely to enter the union as would be predicted on the basis of their percentage in the student population A chimpanzee was trained to make samedifferent judgments about pairs of stimuli For pictures of objects the animal was correct on 23 out of 24 trials Did the chimp perform significantly better than chance From Sprinthall 1990 A researcher is interested in whether or not a significant trend exists regarding the popularity of certain work shifts among police officers A random sample of 60 police officers is selected from a large metropolitan police force The officers are asked to indicate which of three work shifts they preferred The results show that 40 officers prefer the first shift 10 prefer the second shirt and 10 prefer the third shift Do the results deviate significantly from what would be expected due to chance A professor wants to determine whether her department should keep the requirement of college algebra as a prerequisite for an Introductory Statistics course Accordingly she allows some students to register for the course on a passfail basis regardless of whether or not they have had the prerequisite Of the 70 students in the class 40 have had algebra and 30 have not At the end of the semester the professor compares the number of students passing or failing the class with whether or not they had algebra The results are presented blow Are students more likely to pass the course if they have taken college algebra 12 lilil In a study of intraspecif1c aggression aggression directed toward other members of the same species an experimenter nds that 16 out of 23 animals tested in species A exhibit aggression wile only 6 of 25 are aggressive in species B Do the two species differ signi cantly in terms of intraspecific aggression 05 In a study of the effectiveness of an antipsychotic drug patients treated with the drug were compared to patients receiving a placebo In terms of the number relapsing 698 of 1068 patients relapsed after taking the placebo while 639 out of 2127 patients relapsed after taking the antipsychotic drug Test the prediction that the antipsychotic is signi cantly more effective in preventing relapse than the placebo Answers 1 HO Males and females are equally likely to enter the student union H1 Males and females are not equally likely to enter the student union Decision rule If X2 2 384 or isz S 384 reject HO Conclusion sentence Males and females are not equally likely to enter the student union X2 1 N 2000 10368 p lt 05 2 H0 The chimp did not perform significantly better than chance H1 The chimp performed significantly better than chance Decision rule If X2 2 384 or isz S 384 reject HO Conclusion sentence The chimp performed significantly better than chance X2 1 N 24 2016 p lt 05 3 HO Police officers do not have a preference in terms of the shift they work H1 Police officers have a preference in terms of the shift they work Decision rule If X2 2 599 or isz S 599 reject HO Conclusion sentence Police officers have a preference in terms of the shift they work X2 1 N 60 300 p lt05 4 HO Students are not more likely to pass the course if they have taken college algebra H1 Students are more likely to pass the course if they have taken college algebra
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