Chap. 7-13 Notes
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Factorial ANOVA Module 7 Review Comparative Statistics Three uses of Statistics Describe Compare Measure relationships When comparing two groups When comparing more than two groups Comparative Statistics Illustrated Review Critical Designations Samples Independent Related Variables Independent Number Dependent Number Three Kinds of Comparative Tests Two Groups t tests 1 One sample 2 Independent samples 3 Dependent sample gtTw0 Groups ANOVA 1 One way 2 Factorial 3 Repeated measures Review OneWay ANOVA One Independent Variable one way Multiple groups Divided into categories One Dependent Variable Normally distributed Interval data ANOVA Key Terminology Factor Independent or manipulated variable Usually a category or grouping designation Example education level or treatment type Level of the factor Variations in the factor Example HS degree BS degree graduate degree or treatment A vs treatment B vs treatment C Singlefactor design Study involves one factor Factorial design Study involves more than one factor OneWay ANOVA Compares the means of two or more groups of participants on a single FACTOR independent variable m categorical explanatory variable IVs m quantitative response variable DV H0 111 112 11k versus Ha at least one of the 113s differ Practice Exercise 1 and 2 Factorial ANOVA Used with two or more treatment independent variables Each Independent Variable has more than one level Value Assess effects of each independent variable Assess the interaction of each variable How does two variables together work vs each alone Factorial ANOVA 2x2 ANOVA Two independent variables Each has two levels or distinct values Example Teacher effectiveness Experience lt44 Degree minormajor 2x3 ANOVA Two independent variables One variable has two levels other has three 3X3X2 ANOVA TwoFactor ANOVA Used to study two factors simultaneously Uses separate samples for each treatment condition E g effect of heat and humidity on exercise performance Compare mean differences in heat levels Compare mean differences in humidity levels Search for an interaction between combinations specific heat and humidity levels Main Effects amp Interactions Main effect is the comparison of various levels of the same factor e g temperature level F ratios will be calculated for each of the possible main effects Interaction effects are differences that result from a combination of the factors A separate F ratio is calculated for the interaction as well Interactions will be present if The two factors are interdependent change in one depends on change in the other Data of the different factors converge graphically Main effects and interactions are independent of one another Calculation of Fratios Step 1 State the hypothesis Main Effects Interaction Step 2 Locate critical region Set alpha level Step 3 Perform the analysis Step 4 Make the hypothesis decision Main effects Interaction Demonstration Statistics teacher giving an essay final randomly divides the classes in half half the class writes with a blue book and half with computers The students are also designated by typing ability no typing ability some typing ability and highly skilled The dependent measure will be the score on the essay part of the final exam Considerations Dependent Variable Independent variables Levels groups Within variables 2X3 ANOVA Null Hypotheses Descriptive Statistics Table of Means Main Effects Main Effects Differences of one IV across the other IV Method 3033 and 3056 Statistically significant 0 Ability 0 2733 3338 3017 Statistically significant Simple Main Effects The effect of one factor at any given level of the second factor Ability s affect With bluebook 2667 31 3333 Method s effects at any ability 3333 27 Interaction Effects The change in the main effect of one variable over levels of the other unaccounted for by the two variables alone ANOVA PRINTOUT Dependert re39ieele i Te eef etueenSuhjeeteEffede Type III Neneerrt Deeerrr 5Lrn ef Meen Perernet eel Seuree Squares elf SE1 Llere F 5ig er F39errtera Cerreetezl 141 ejel 2334443 5 425 344 53 12232 231 Intercept 1553355 1 1553355 12E53 533 12II5343 1333 AEIIIJT r 122444 2 53222 4535 533 3213 554 141 ETHDD 222 1 222 315 331 315 352 AEIIIJTH WHEI39HDD 113223 2 553 4334 34 3353 533 Errer 155533 12 13333 Tetel 1233353 13 Cerreetezl Tetel 434444 1 El Cemented Lle39ng Emma 35 lit H Eiquereel 533 AdjueteUH Squared 4131 FACTORIAL ANOVA USING PASWSPSS ANALYZE GENERAL LINEAR MODEL Independent Variables entered into FIXED FACTORS OPTIONS DESCRIPTIVES Practice Analysis Exercises 3 through TASK EXERCISE 7 672 Week 9 Repeated Measures and Analysis of Covariance Review ANOVA Compares variability Within each group to variability across groups Null Hypothesis Three Kinds of Post Hoc Tests Review Factorial ANOVA How Many Independent Variables Factors Levels Main Effects Interaction Effects Practice Worksheet ANCOVA Exercise 1 Exercise 2 TASK EXERCISE 7 Factorial ANOVA Repeated Measures ANOVA Compares means across a single factor Uses a single sample and all subjects are measured using all treatment conditions Example Measure depression before therapy after therapy and six months after end of therapy Advantages Limited subject supply Eliminates effect of individual differences Disadvantages Testing effect WHY USE REPEATED MEASURES DESIGN Individual differences can be eliminated or reduced as a source of between group differences Sample size is not divided between conditions or groups tests are more powerful Economical when sample members are difficult to recruit Differs From Independent ANOVA The scores for the dependent variable appear across multiple columns Example Sleep Deprivation and Motor Skill OneWay Repeated Measure ANOVA Analyze GLM Repeated Measures Enter the name of the first factor Need to Name the factor Tell the computer how many levels there are What they are DEFINE Options Right now just descriptive statistics Example Repeated Measures ANOVA Does exercise affect heart disease by reducing in ammation Might this protection be gained over a short period of time 20 participants 6 month exercise pgm Measured the in ammatory marker called CRP pre training 2 weeks into training and post 6 months training Independent Variable Dependent Variable Related or Independent sample Levels Example Output Descriptive Statistics Mean Std Deviatien N PreTraining 99925 92259 29 Weelc2 29525 59955 25 F39estTraining 22459 49592 29 Within Subject Effects Tests efWithinSuhjects Effects MeasureSSP Seurce Type III Sum Partial Eta enguares df Mean Sguare F Sig Sguared time Spnericitgr lssumed 5959 2 4154 21992 955 525 I SreenneuseSeisser 5999 1121 2992 21592 959 525 HuynnFeldt 5959 1252 5912 21992 955 525 Lewer peund 5959 1955 9955 21992 955 525 Errarltime Spnericitgr lssumed 2595 SS 195 SreenneuseSeisser 2595 22252 992 HuynnFeldt 2555 22592 929 Lewer peund 2595 19599 995 Pairwise Comparison Pairwise Cemparisens Mea s ureCHF39 I time 51 time 9555 Sen dence lntenral fer IIEIi1 lerencea Mean Difference l cli Std Errer Siga Lewer Seund Upper Seund 1 2 125 952 149 999 229 9 542 129 999 959 1295 2 1 125 952 149 229 599 9 229 152 991 299 1142 S 1 S42 1 TS 955 1295 999 2 43929 152 991 1142 295 Sased an estimated marginal means a Adjustmentfer multiple cemparisens Senferreni The mean difference is signi cant attne 95 level Analysis of Covariance ANCOVA 0 An extension of ANOVA tests effects of IVs on a single DV 0 Can be used to account for variables that have a known effect on the DV but are not IV 0 E g studying the effect of education on job satisfaction but want to account for gender 0 These variables are called covariates or intervening variables and can be controlled 0 These covariates are controlled for and the analysis mirrors a standard ANOVA 0 Main effects amp interactions are assessed after the effect of the covariates have been removed Purposes of ANCOVA 0 Increase sensitivity of F tests 0 Removes predictable variance from the error term 0 Statistical adjustment to account for non random subject selection 0 Interpret differences in levels of IV when measuring several DVs Choosing covariates 0 Variables that affect or have the potential to affect the DV 0 Number of covariates depends on 0 Known relationships based upon previous research 0 Number of IV levels or groups 0 Total number of subjects Assumptions amp Limitations 0 Samples are randomly assigned 0 Score distributions are normal compared to the population 0 Sample variance is similar to population variance 0 A direct relationship exists between DV and covariates 0 Covariate has stable variance 0 Covariate is measured without error Hypothesis Testing with ANCOVA Step 1 Determine hypothesis and alpha level Step 2 Determine critical region for hypothesis decision Step 3 Gather and analyze data Test of sample distributions Test of sample variance Examine interaction Examine main effects Step 4 Make hypothesis decision USING SPSS TO RUN ANCOVA Removes the effect of a known covariate Covariate and DV are at least interval data Demonstration 0 Using weekend 1 examples 0 T test data 0 Test for differences in performance 0 Across teachers 0 For required and elective Practice Worksheet Ancova New Exercise 1 New Exercise 2 TASK EXERCISE 8 ANCOVA PES 672 Week 10 Relational Statistics Using Correlation and Regression Review of Comparative Statistics Critical Issues 0 How many groups are being compared 0 What isare your Independent variables 0 How many levels are there 0 What IS THE dependent variable 0 Are your samples related or independent Comparative Tests in Our Toolbox Ttests One sample 0 Related Samples 0 Independent Samples Analysis of Variance 0 Independent samples 0 Related samples 0 Poor controls Conditions Necessary to Use Parametric Statistics Random sampling Necessary and sufficient sample sizes Dependent variable must be interval Normal distributions or equal variances Practice Handout Exercise 1 Automobile Gas mileage Worksheet Exercise 1 Correlation 0 Single number that describes the degree of relationship between TWO variables 0 Most often used when a phenomenon of interest cannot be tested directly 0 Study indirectly by observing one variable and observing the frequency or strength of the second variable naturally Example Height and SelfEsteem Can you conduct a test Measure both in single subjects and observe the relationship Null Hypothesis There is no relationship Alternate Hypothesis 0 A relationship exists Regression Attempt at modeling a phenomena that exists between one scalar variable and one OR MORE explanatory variables Relationship is described in a LINEAR equation that best describes the relationship between the variables Example Y2a3b6 Uses of Correlations Prediction Does a change in one variable coincide with change in another 0 If one variable is at a certain level is the value of another variable predictable or follow a pattern Validity One variable measures a defined construct does another variable measure the sameopposite construct Reliability 0 Are measurements of a variable consistent Theory Verification Based upon observation of apparently linked variables or events can we qualify and quantify that link Example Data Relationship between Cigarettes and Lung Cancer gt Relationship between a quantitative response variable and a quantitative explanatory variable gt Observations from 11 countries 0 The explanatory X variable 0 per capita cigarette consumption in 1930 CIG1930 0 The response Y variable 0 lung cancer mortality per 100000 LUNGCA Use of the Scatterplot Bivariate xy data points are plotted to form a scatterplot Make certain the explanatory x variable is on the horizontal axis The following elements are inspected 0 Form Can the relation be described with a straight or other type of line 0 Direction Do points tend to trend upward or downward Strength of association Do point adhere closely to an imaginary trend line Outliers if any Are there any striking deviations from the overall pattern Correlational Strength gt The degree to which points adhere to a trend line gt Eye is not a good judge of correlational strength gt Correlational strength quantified via a correlation coef cient Pearson Product Moment Correlation Coef cient r gt Quantifies a linear relationship between X amp Y gt Always falls between 1 and 1 gt r1 all data points fall on a line with upward slope gt r 1 All data points fall on a downward slope gt Positive relationship gt Upward trend gt Negative relationship gt Downward trend gt Strength of relationship gt The closer r is to 1 or 1 Calculating r z scores quantify the distance a value lies above or below the distribution s mean gt When z scores for X and Y track in the same direction their products are positive gt When z scores for X and Y track in opposite directions their products are negative gt Thus the correlation coefficient tracks the degree to which X and Y go together Interpreting the Correlation Coef cient r lt20 r 20 to 39 r 40 to 59 r 60 to 79 r 80 to 100 Characteristics of Correlations Direction Positive r gt 0 negative r lt 0 or no association r z 0 Strength The closer r is to 1 or 1 the stronger the association Reversible relationship 0 X or Y can be the explanatory variable calculations come out the same either way Coef cient of determination The square of the correlation coefficient r2 Quantifies the proportion of the variance in Y explained by X Linear relations only Correlation applies only to linear relationships Correlation VS causation Lurking variables may explain statistical correlations Problems with Outliers Can have a profound effect on r The figure has an r of 082 that is fully accounted for by the single outlier Example Height and SelfEsteem Can you conduct a test Measure both in single subjects and observe the relationship Null Hypothesis There is no relationship Alternate Hypothesis 0 A relationship exists Height and Selfesteem 0 Suppose we hypothesis that how tall you are effects your self esteem Converse 0 Collect some information on twenty males in inches 0 Self esteem is measured based on the average of 10 1to5 rating items higher scores mean higher self esteem Running the Statistical Test Testing the Signi cance of a Correlation WHY What does it tell you Why Hypothesis Test To guard against declaring too many random correlations as significant Measure Strengths of LINEAR Relationships autosales PEARSON PRODUCT Variables Only linear relationships Normally distributed Conditions for Inference Independent observations A bivariate sampling distribution of X and Y 0 Bivariate Normality is especially important when making inferences about strong correlations 0 r can still be used descriptively when data are not Normal Spearman Correlation Measures degree of relationship between two variables Used for variables that are ordinal or nominal data 0 have a curvilinear relationship 0 Are not normally distributed Regression Line Using a least squares method we find the best fitting line that describes the relationship between X and Y This method works by minimizing the sum of squared residuals A residual is the distance of a point from the line The linear model is y abx Where y E a predicted value of Y a E the intercept of the line b E the slope of the line Conditions for Inference Inference about regression analysis conditions that can be remembered with the mnemonic LINE Linearity Independent observations ormality at each level of X Equal variance at each level of X TASK EXERCISE 9 Correlations And I am positive you can do it PES 672 Week 18 Linear Regression Correlations De ned A statistical measurement of the relationship between two variables gt Value between 1 and 1 gt Strength of relationship gt Direction of relationship Linear Regression De ned Mathematical technique for nding the straight line that best ts the values of a linear function plotted on a scatter graph as data points Used as the basis for estimating the future values of the function Correlation vs Regression Scatterplots and Correlation A scatter plot is used to show the relationship between two variables Correlation analysis is used to measure strength of the association linear relationship between two variables Only concerned with strength of the relationship No causal effect is implied Scatter Plot Examples Linear relationships Curvilinear relationships V 39 y x x ll 3 Correlation Coef cient Features Unit free Range between 1 and 1 The closer to 1 the stronger the negative linear relationship The closer to 1 the stronger the positive linear relationship The closer to O the weaker the linear relationship Examples of Approximate r Values y y Y Hera 1 a H x x x rE l rE6 r20 V g y 23 g r39 ri3 x r1 x Conceptual Example Tree Height Is there a relationship between how wide a tree is and how tall it is If so can we use the diameter of a tree to estimate predict its height Tree DiameterHeight Data and Scatterplot TREEHEIGHT 2 5 2 y 597360214551 TREE DIAMETERX 8 9 7 6 13 7 11 12 Tree 70 Height y 60 O 50 O C 40 30 O 20 0 10 0 0 2 4 5 a 10 12 14 Trunk Diameter 1 Calculating a Correlation Coef cient Interval Data Analyze Correlation Bivariate OUTPUT Correlations height diam height Pearson Correlation 1 787 Sig Zrtailed 020 N 8 8 diam Pearson Correlation 787 1 Sig Zrtailed 020 N 8 8 Correlation is signi cant at the 005 lievel 2 tailed Signi cance Test for the Correlation Hypotheses H0 p 0 no correlation HA p 7 O correlation exists Conclusion There is evidence of a linear relationship at the 5 level of signi cance Introduction to Regression Analysis Uses Predict the value of a dependent variable based on the value of an independent variable Explain the impact of changes in an independent variable on the dependent variable Dependent variable variable we wish to explain Independent variable variable used to explain the dependent variable Simple Linear Regression Model Only one independent variable x Relationship between x and y is described by a linear function Changes in y are assumed to be caused by changes in x Linear Regression Equation Population Linear Regression SOOOO Regression Model Interpretation of the Slope and the Intercept gt be is the estimated average value of y when the value of x is zero gt b1 is the estimated change in the average value of y as a result of a oneunit change in x Calculating a Linear Regression Model with SPSS Analyze Regression Linear Enter independent and dependent variables Choose descriptives if interested Use Coef cients table to construct the Equation Constant Under B within Unstandardized Coef cients Slope Under B with Independent variable Practice Use Tree Diameter Data Coefficientsa Model Sta ndard iized U nstandtardized Coef cients Coef cients B Std Error Beta t Sig 1 Constant 3261 14365 22 828 diam 4755 1522 78 3t 25 020 a Dependent Variable height Slope Intercept Computer Exercise 10 Linear Regression Comparative Statistics Nonparametric data Using Chisquare Review Relational Statistics Analyzes the strength of relationship between two variables What does the pvalue tell us T tests for three situations Samples measured with interval scales Samples measured with nominal or ordinal scales When do we use Pearson39s Product Spearman39s rho Review Comparative Statistics Marriage between Statistics and Probability Levels of signi cance pvalues Do our samples behave the way we expect them to Extremely low signi cant values mean again the signi cance level tells us Review ttests Compares two groups Null hypothesis Assumptions Random sampling Dependent variable is interval data Samples come from a normal distribution WHY Review ANOVAs Compares more than two groups at one time Can compare one or more independent variables Null hypothesis Assumptions Dependent variable is measured on an interval scale Independent variables can be categorized Distributions are normal The Basis for Parametric Statistics Why do the measurement scales of ttests and ANOVAs have to be interval Why must the population scores be normally distributed Factorial ANOVA Looks as multiple independent variables at one time MAIN EFFECTS Are there differences between groups in each of the factors as a whole INTERACTION EFFECTS Is there a difference in how groups of one factor responds to the second factor Practice Exercise 1 Quality Control Analyze Does the number of coats of lacquer make a difference after a point Does it matter what kind of bead is used Do different beads need different numbers of coats Practice Exercise 2 School Data Analyze What is the relationship between kicking and dribbling skills Does knowing the score on one test help us predict the performance on the other If so to what degree Nonparametric statistics What the ampamp Are NonParametric Statistics Used when we can not assume that our data will be normally distributed No mean or expected variance between scores Ordinal and nominal data Instead of comparing to a normal distribution compare to expected proportions Differences in Tests T test Compares the sample means relative position on the assumed normal curve What are the odds that the values we got happened by random chance ANOVA Compares differences in variances between scores and across scores What are the odds that the difference in variance occurred by random chance Chisquare Explained Since we have no continuity in measure scales no order Since we have no probability in terms of normal distributions Calculates the odds that the distribution of scores matches what we would expect within the probability of chance Chi Square Test X2 Discussed Divides observations into groups called quotCELLSquot Compares observed frequencies with expected frequencies Nominal or ordinal data Categorical responses Differences in the frequencies of responses vs proportions Not as sensitive in detecting differences Does not use mean amp SD to describe a population Proportions amp frequencies Example Views on Gun Control Suppose a researcher is interested in voting preferences on gun control issues A questionnaire was developed and sent to a random sample of 90 voters The researcher also collects information about the political party membership of the sample of 90 respondents Results Gun Control Overall over 90 respondents 25 favored gun control 40 opposed gun control 25 were neutral on the issue Question Did the opinions differ by party lines Bivariate Frequency Table or Contingency Table Expected Favor Neutral Oppose f row Frequency In Red Democrat 50 Republic an 40 f 90 column 1 Determine Appropriate Test 1 Party Membership gt 2 levels and Nominal 2 Voting Preference gt 3 levels and Nominal 3 Determine The Hypothesis Ho There is no difference between D amp R in their opinion on gun control issue Ha There is an association between responses to the gun control survey and the party membership in the population 2 Establish Level of Signi cance Alpha of 05 4 Calculating Test Statistics SPSS Output for Gun Control Example ChiSquare Tests Asymp Sig Value df 2sided Pearson ChiSquar 11025a 2 004 Likelihood Ratio 11365 2 003 l l mammal 1 N of Valid Cases 90 8 0 cells 0 have expected count less than 5 The minimum expected count is 1111 Decision on Gun Control Test statistic 1103 exceeds critical value Null hypothesis is rejected Democrats amp Republicans differ signi cantly in their opinions on gun control issues Three Types of Chisquare Tests Goodness of Fit Tests to see if an observed distribution differs from a theoretical distribution Tests of Independence Assesses whether paired observations on two variables expressed in a contingency table are independent of each other Tests of Homogeniety Example of Goodness of Fit Gym assumes highest attendance on Mondays Tuesdays and Saturdays lowest on Friday and Sunday Compare assumed attendance to actual Example of quotGoodness of Fitquot Is a die honest One die 100 rolls How many 1395 2395 3395 etc Is a die honest One die 100 rolls How many 1395 2395 3395 etc Example of Test for Independence Is there a gender gap in political ideologies Sample 1000 voters Divide them into quotover 40quot and quotimmaturequot AGE GAP Calculating a chisquare Test for Independence SPSS Procedure Analyze Descriptive Statistics Crosstabs Dependent Variable in Rows Independent Variable in Columns Display clustered bar charts Statistics Chisquare Continue Cells List observed and expected if you are interested Comparative Statistics Nonparametric data Using Chisquare Nonparametric statistics What the ampamp Are NonParametric Statistics Used when we can not assume that our data will be normally distributed No mean or expected variance between scores Ordinal and nominal data Instead of comparing to a normal distribution compare to expected proportions Chisquare Explained Calculates the odds that the distribution of scores matches what we would expect within the probability of chance Null Hypothesis The distribution of scores across cells are the same for subjects in all group Alternative Hypothesis The distribution of scores across cells differ for subjects in at least one group Chi Square Test X2 Discussed Divides observations into groups called quotCELLSquot Compares observed frequencies with expected frequencies Nominal or ordinal data Categorical responses Differences in the frequencies of responses vs proportions Three Types of Chisquare Tests Goodness of Fit Tests to see if an observed distribution differs from a theoretical distribution Tests of Independence Assesses whether paired observations on two variables expressed in a contingency table are independent of each other Tests of Homogeniety Example of Test for Independence Is there a gender gap in political ideologies Sample 1000 voters Divide them into quotover 40quot and quotimmaturequot ChiSquare Test of Independence Borderline Personality Disorder ls diagnosis of borderline personality disorder independent of gender among patients demonstrating suicidal behaviors 200 patients demonstrating suicidal behaviors with diagnosis Review Exercise 1 672 dx Status Calculating a chisquare Test Goodness of Fit SPSS ChiSquare Goodness of Fit Do the observations we drew randomly match what we would have expected One sample compared to an expectation One sample compared to a second sample How do we designate the expected distribution Weigh each probability equally Use past or present trends Example of quotGoodness of Fitquot Is a die honest One die 100 rolls How many 1395 2395 3395 etc Issue One What will SPSS Use to set Expectations Choose Data Weight Cases Select the variable used to set expectations Run SPSS Goodness of Fit Analyze Nonparametic tests Legacy Dialogs Chisquare Choose variable of interest ChiSquare Goodness of Fit Hospital Discharge data Large hospital schedules discharge support staff Assumes patients leave at a constant rate through the week Wonders if this is true dischargedata Hypothesis Testing with X2 Test for Goodness of Fit Null Hypothesis Alternative Hypothesis Independent Variable Dependent Variable Measure Scale Goodness of Fit Hospital Discharge 1 What variable constitutes the cases 2 Tell the computer which variable will be weighted day Discharge admits De nes the expected distribution Data Weight cases 1 Analyze using Goodness of Fit Assume all cells equal Frequency Chart Output Statistics Table Sig the probability of obtaining a chisquare value greater than or equal to 29389 if patients are discharged evenly across the week Conclusion Interpreting Chisquare Signi cance plt critical value The observed data was distributed signi cantly differently than what was expected by chance alone This leads us to reject the null hypothesis and accept the fact that the observed frequency is signi cantly different than the norm Pgt critical value No signi cant deviation occurred in the observed data from what was expected at plt005 Hospital Data What about Weekdays Necessary Assumptions Can only be used on data with the following characteristics 1 The frequency data must have a precise numerical value and must be organised into categories or groups The data must be in the form of frequencies The expected frequency in any cell must be greater than 5 The total number of observations must be greater than 20 weww Types of Chisquare Tests 1 Is there a signi cant difference in the proportions of various subgroups gt Goodness of Fit 2 Are two variables independent ls af uence related to or independent of motivation to work Test of Independence 3 Are subgroups homogeneous Is the distribution of a speci c characteristic similar across groups ChiSquare Goodness of Fit Customizing Distributions Clothing manufacturer tries rstclass postage for direct mailings hoping for faster responses than with bulk mail Ordertakers record how many weeks after the mailing each order is taken Exercise 3 mailresponse Hypothesis Testing with X2 Test for Goodness of Fit Null Hypothesis Alternative Hypothesis Independent Variable Dependent Variable Measure Scale Goodness of Fit Mail Response 1 What variable constitutes the cases 2 Tell the computer which variable will be weighted week First Class Bulk De nes the expected distribution let39s use rst class Data Weight cases 1 Analyze using Goodness of Fit Assume all cells equal NO Frequency Chart Output Statistics Table Sig the probability of obtaining a chisquare value greater than or equal to 1235 if patients are discharged evenly across the week Conclusion TASK EXERCISE 12 Chisquare Goodness of Fit Comparative Statistics Nonparametric data Ordinal Data Mann Whitney Review NONParametric Statistics Focus on scale of the dependent variable Used when there is no equal intervals between scores No normal distributions Not as powerful as parametric statistics Comparing Dependent variable is nominal Chisquare Comparing Dependent variable is ordinal Mann Whitney 1 IV or Kruskal Wallace gt1 IV Review Categorical Nominal data Dependent variable is nominal Measures of central tendency no longer make sense Chisquare Goodness of Fit Compare the set of data to an expected model SPSS Chisquare Test of Independence Tests whether two independent samples are from the same distribution 0 Dependent Variable Classification of Group membership SPSS Analyze descriptive statistics crosstabs IV in Rows DV in Columns Check Chisquare Practice Chisquare Fast Food Consumption Data from the STARS project Telephone survey investigating people39s quotfast foodquot eating habits 400 people personal characteristics and purchasing behavior Do people from different parts of the country differ in fast food consumption Practice Chisquare Null Hypothesis Dependent Variable Scale Independent Variable Number and levels Test Critical Value Frequency Chart Output Statistics Table How is ChiSquare Goodness of Fit Different Compare the set of data to an expected model You have number of occurrences in each cell vs individual scores of each subject First weigh cases by dependent variable data weigh cases Analyze Nonparametrics Legacy Dialogue Chisquare Practice Chisquare Does What shisname your stats professor give equal numbers of A s B s C s D s and F s as final grades in his graduate statistics classes The observed frequencies are A 6 B 24 C 50 D 10 F 10 Null Hypothesis Dependent Variable Scale Independent Variable Number and levels Test Critical Value Practice Chisquare Results Results Analysisconclusion Practice Chisquare Results What if the normal distribution of grades in years past was as follows 10 30 40 10 10 Is this year different munwgt ANALYZING ORDINAL DATA Ordinal Data Techniques Ordinal data or data that uses ranks can not have a true mean or SD calculated Can not use traditional descriptive or inferential statistics Can be used to measure less objectivequantitative measures Can convert interval or ratio data into ordinal Simpler When assumptions are violated Undetermined score Ordinal Data Tests MannWhitney U Test Uses data from 2 separate samples to evaluate 2 treatments or populations Alternative to independent ttest Wilcoxon Test Repeated measures between two treatment conditions Alternative to repeated measures ttest KruskalWallis Test Uses data from 3 or more samples to evaluate 3 or more treatments or populations Alternative to single factor ANOVA MannWhitney U Test Used to evaluate the differences between two treatments or populations ordinal ttest ORDINAL DATA Independent ttest Works with the following logic A real difference between groups One set of scores to have a greater concentration at either end of the ranking No difference between groups Even distribution of scores from both sets throughout the ranking Calculation of MannWhitney s U A separate sample is obtained from each of the treatments Samples are combined and then ranked A numerical value is assigned to each score A score gets a point for each score that it outranks The points for scores from each sample are totaled and the lower of the two sample totals is the U value Example Data Group A Scores Group B Scores 27 18 2 28 6 30 9 3 1 15 8 Example of U Score Calculation Rank Score Sample Points for Each 1 2 A 9 2 6 A 8 3 8 B 7 4 9 A 6 5 15 A 5 6 18 B 4 7 27 A 3 8 28 B 2 9 30 B 1 10 3 l B 0 Sum the scores for each group UA 31 UB 14 SPSS MannWhitney s U Tests Whether two independent samples are from the same distribution Dependent Variable Classification of Group membership Example MannWhitney s U Practice Mann Whitney Physical Therapy Null Hypothesis Dependent Variable Scale Independent Variable Number and levels Test Critical Value DATA ADL in Blackboard Folder ADL Results Mann Whitney U Results Practice Mann Whitney Mode of Instruction A university wondered if students benefited as much or more from a computerbased instruction rather than the traditional lecture format when learning statistics Practice Mann Whitney Physical Therapy Null Hypothesis Dependent Variable Scale Independent Variable Number and levels Test Critical Value Use InstructMode data in Blackboard Results Mode of Instruction TASK EXERCISE 12 Mann Whitney U SAMPLE CASES Example 1 Are male adolescents who do not engage in sports more depressed than the average male adolescent 30 adolescent boys who have indicated they do not engage in sports Test a depression measure that was standardized to have a mean of 50 for male adolescents A value less than 50 implies less depression than a typical male adolescent A value greater than 50 implies more depression than a typical male adolescent 20 Sports Science students selected from the population to investigate if a 12 week plyometric training program improves their standing long jump performance In order to test whether this training improves performance the sample group are tested for their long jump performance undertake a plyometric training program and then measured again at the end of the program Example 2 Person with too much free time wonders if a football filled with helium travels farther on average than a football filled with air 18 adult male volunteers randomly divided into two groups Group I kicked a football filled with helium to the recommended pressure Group 2 kicked a football filled with air to the recommended pressure Distances measured in yardage Example 3 Researcher predicts students will learn best with a constant background sound as opposed to unpredictable or no sound Randomly divides twentyfour students into three groups All students study a passage of text for 30 minutes Group 1 background sound at a constant volume Group 2 study with noise that changes volume periodically Group 3 study with no sound at all After studying all students take a 10 point multiple choice test over the material Example 4 Department of computer science s admissions committee wants to know if there is a relationship between Computer Science GPA and SAT taken as a high school senior Five college students39 have the following rankings in math and science courses Example 5 10 singers are evaluated by two judges Judges rank the performers by performance Are the rankings by the judges Virtually the same or significantly different