Study Guide Chapters 1 -- 24 (and Which Significance Test in Which Situation) Social Work Statistics 2223 2014 -- Rosenthal General Hints for Tests: Problems and Questions at back of each chapter are best guide to types of problems and questions. All test questions are multiple choice and true false. As an estimate, perhaps 30% of the questions on the test involve computations and about 70% do not. Formulas will be provided but you will need to know symbols. Though this is “the” study guide, the “Chapter Summary” at the end of each chapter also functions, basically, as a study guide – much the same information in these two places Chapter 1 more questions on the four levels of measurement than on anything else in Chapter 1 what are these levels? what are the key characteristics of each level? what conclusions can and cannot be drawn for variables at different levels? recognize variables at different levels of measurement B what level of measurement is a rank ordering at? (ordinal) basics of evidenced-based practice quantitative vs. qualitative research various definitions: statistic, case, constant, etc. categorical/qualitative vs. numeric/quantitative variables recognize variables (e.g. eye color) versus values (e.g., blue, brown, hazel) independent and dependent variables two major branches of statistics: descriptive and inferential recognize basic purposes of each descriptive ???? describe study sample inferential ???? draw conclusions about population from which sample randomly selected univariate (one) vs. bivariate (two = relationship) vs. multivariate (three or more) statistics confounding variable can you recognize a random sample (methods of chance used to select) from a nonrandom one? valid uses of inferential statistics – must have random sample can only generalize to population from which that sample selected. can you recognize when a study uses random assignment to groups? = methods of chance used to assign participants to groups random assignment eliminates confounding variables helps researcher draw conclusions about cause1 fact of relationship or difference in (random) sample does not necessarily indicate that relationship/difference also exists in population difference can occur due to the luck of the draw = chance statistical significance tests assess the likelihood that a difference/relationship is due to chance. Chapter 2 frequency vs. sample size know your symbols: f N etc can you read a simple frequency distribution table and determine frequencies, cumulative frequencies, and cumulative percentages? perhaps 1 or 2 question on figures -- can you distinguish between the different types of figures? percentile (percentile rank) ???? percentage of cases with lower values when is a bar chart used? (categorical data) when is a histogram used? (numeric data) when is order of bars arbitrary ???? nominal data; not arbitrary, however, when ordinal data what do the various parts of a box plot convey? e.g. line in middle of box ???? 50th percentile pie chart ???? slice of pie proportionate to percentage/proportion of cases in category Chapter 3 can you carry out simple calculations of mean, median, and mode? Note: no formula is provided for mode or median do you know basic definitions of mean, median, and mode? advantages and disadvantages of each measure given interval/ratio-level, which measure is most commonly used -- mean at what levels of measurement can each measure of central tendency be used? mode = all levels; median = ordinal and interval/ratio; mean = interval/ratio you might get a trick question right that asks you: what is the median eye color or similar? a nominal-level variable has no median ordinal-level categorical variable ????median value with 1st cumulative percentage ≥ 50% when is the median preferred to mean (when there are outliers – recognize outliers in data) in which direction do positive (upwards) and negative outliers (downwards) pull the mean trimmed mean Chapter 4 basic concept/definition of variability = how dispersed/spread out are observations what percentiles are involved in calculation of the interquartile range? (25 and 75) which measure of variability is the most common? (standard deviation) which distribution has greater variability?: i.e. “2,3,3,3,4" vs. “1,2,3,4,5" (the second one) variability for categorical variables as in, for instance, Table 4.1 nearly all in one category ???? low variability equal or nearly equal distribution in different categories ???? high variability can you calculate a deviation score from the mean? e.g. Mean = 8, Case = 5: 5 - 8 = -3 can you calculate a mean deviation? can you calculate a standard deviation with very simple data (3 or 4 cases)2 what happens to the standard deviation a constant is added to each score? (it is unchanged) the standard deviation equals the square root of the variance (s2 = 100 ???? s = 10) the variance equals the standard deviation squared (s = 10 ???? s2 = 100 ) Chapter 5 are all variables normally distributed? B no is the normal distribution a theoretical (abstract/mathematical) distribution or a Areal@ one? B theoretical do any real world variables have a precisely normal shape? no percent of cases with 1 SD of mean = 68% (within 2 SDs = about 95%) percent of cases below/above the mean in a normal distribution? (50%) can you do problems about (for normal distribution): percentage of cases 1, 2, and 3 SDs above and below the mean? more than 1, 2, 3 SDs away from mean, greater than or less than given SDs, etc) (will have access to figure 5.12) do you know that you cannot do problems involving percentages when the distribution is nonnormal in shape? skewness = departure from symmetry recognize positively and negatively skewed distributions (tail tells tale) positive skew ???? cases cluster on negative (left) side; positive (right) tail elongated negative skew ???? cases cluster on positive (right) side; negative (left) tail elongated skewness can have important implications for which statistical procedures can be used can you reason out using Figure 5.8 (p 56) which vertical lines convey which measures of central tendency? kurtosis = peakedness ???? less important than skewness regardomgappropriate procedures flat and bimodal distributions for a straightforward example, can you picture the shape of distribution? For instance, what is the shape of the distribution of family income in the USA [positively skewed] z score = how many SDs a score is above or below the mean be able use the z score formula: X = 20, X̅= 15, s = 10: z = (20 - 15)/10 = 0.50 using (provided portions of) the normal distribution table, figure percentages of cases above, below, within, more extreme than, percentile rank, etc. given z scores (assuming normal distribution) for all z scores: 0 = mean, 1 = SD (negative z scores are below mean; positives are above) in this regard, z scores are useful even for nonnormally distributed variables know: z scores may be calculated for all numeric variables problems involving percents can only be done if distribution has normal shape Fred=s score is 1.22 SD=s below the mean. What is his z score? (-1.22) Fred=s z score is -1.22. How many SDs is his score above or below the mean? (1.22 below) Chapter 6 know basic definitions of relationship (relationship/association = synonyms) certain values of one variable tend to “go with” (occur more often with) certain values of the other3 (T or F) All relationships are causal -- False recognize that relationship conveys pattern B need not be a “perfect” pattern for categorical variables: relationship conveyed by differences in percentages 70% of women vs. 50% of men eat drink diet soda ???? relationship as %s differ 57% of Republicans support X and 43% do not? What is the strength of this relationship? nonsense question as there is only one variable involved recognize when variables in crosstab table are (%s differ) or are not associated (%s same) recognize column and row variables and percents, cell frequencies, margins, N, etc. recognize whether cell percentages are column or row percentages excepting 1st crosstab table (table 6.1), text always presents column percentages only be able to calculate D% size/strength of association/relationship and effect size ???? in essence, synonyms D% < 7% ???? small association > 45% ???? very large association qualitative descriptors: interpret size of association in context of social science research rather than in strict mathematical way descriptors for D% only recommended when both %s between 10 and 90 be able to calculate a RR recommendation for RR: use %s close to 0 not those close to 100 if don’t do so, RR will be arithmetically correct but not intuitive/useful reciprocal RRs convey same size of association descriptors for RR should only be used when both %s less than about 30 know when RR tends to be preferred (when one or more percents close to 0%) and when D% is preferred (no percents close to 0 or 100) do remember that you don=t need a crosstabs table to assess whether there is relationship just need to know whether percents differ (no relationship) or are the same (relationship) Chapter 7 odds and odds ratio (calculate: recognize that a problem might provide numbers or, on the other hand, it might provide proportions/percentages -- you should be able to do either) remember: total number in group is not used in calculating odds instead one uses number who experience event and number who do not odds and odds ratio may vary from 0.00 to positive infinity be sure you can distinguish between odds and odds ratio: be able to calculate each odds used in calculating odds ratio odds ratio of 1.00 ???? equal odds of event in both groups odds ratio > 1.00 ???? greater odds in Group 1 than Group 2 odds ratio < 1.00 ???? odds of event lower in Group 1 than in Group 2 odds ratio: the closer to 1.00, the smaller the size of association; the closer to 0 or to positive infinity the greater the size of association Odds ratios that are reciprocals convey the same size of association odds ratio of 1.50 (or 0.67 = reciprocal) ???? small association odds ratio >10.0 (or < 0.10) ???? very large association Which odds ratio 7.7 or 4.4 conveys the stronger association? (7.7) Which odds ratio 0.40 or 0.10 conveys the stronger association? (0.10)4 Which odds ratio 5.00 or 0.20 conveys the stronger association? (same size as are reciprocals) Does an odds ratio of −0.30 convey a small or large association? (trick question, negative odds ratios are not allowed B no such thing) advantage of odds ratio over D% and RR its descriptors may be used to assess size of essentially all associations involving two dichotomous variables descriptors for D% should not be used if a % is < 10 or > 90 descriptors for RR should not be used if a % > 30 Cramer=s V: a preferred measure of nondirectional association for crosstab tables larger than 2x2 OK but not preferred for directional association Directional association vs. nondirectional association both variables must be at least at ordinal level to have directional association positive directional association (as one goes up, so does the other) vs. … negative directional association (values change in opposite directions) Recognize the measures for ordinal relationships --gamma, tau-b, tau-c, Somer=s D-- all may vary from -1.00 to 1.00 Chapter 8 r = Pearson correlation coefficient r may vary from -1.00 to +1.00 the closer the absolute value to 1.00, the larger the correlation if values of variables change in same direction, correlation is positive; if values change in opposite directions, correlation is negative .10 or -.10 ???? small correlation/relationship .50 or -.50 ???? large correlation/relationship r measures degree of linear (straight-line) association when an association is curvilinear, r is not an appropriate measure curvilinear association does an r equal to 0.00 always convey the absence of association? no, it conveys absence of linear association; it does not rule out possibility of curvilinear (or other nonlinear) association have some proficiency in reading a scatterplot slope lower left to upper right ???? positive correlation slope upper left to lower right ???? negative association recognize perfect correlations and the buckshot pattern also (p 112) the more the markers/dots line up in “tight” oval ???? stronger the correlation r2 = coefficient of determination proportion of variance in one variable shared with/explained by another to assess strength of association with Table 8.2 descriptors ???? take square root of r2 adding a constant to one or both variables does not affect the value of r multiplying one or both variables by a constant does not affect the value of r which correlation .4 or -.7 conveys the stronger association? (-.7) r = 3.87: large, medium or small relationship? (trick question, r varies between -1.00 and 1.00) r is a standardized coefficient; B is an unstandardized coefficient r conveys change in terms of standard deviation units (z scores);5 B does so in terms of original units of measure (raw scores) be able use both standardized and unstandardized regression equations be able to do problems like Q19 andQ20 be able to do problems like Q30 know that B is the (unstandardized) regression coefficient and that it conveys (1) the slope of the regression line and (2) the change in predicted Y as X increases by 1.00 know that A is the constant and that it conveys the expected value of Y when X = 0 does r convey size/strength of (linear) association? yes, as it conveys predicted change in terms of z scores does B convey size/strength of (linear) association? no, as it conveys predicted change in terms of original units of measure (“raw” scores) r should be used cautiously (or not at all) when the variability of one or both variables is very low in this situation, that Avariable/s@ is/are very nearly a constant observed correlation will be smaller than (closer to 0.00 than) the correlation that would otherwise be observed Chapter 9 calculate the SMD using the Chapter=s formula -.20 or .20 ???? small difference/association; -0.80 or +0.80 ???? large difference/association Graphical visuals (fig 9.1, p129) the less the overlap between distributions ???? the greater the size of the SMD the more the overlap ???? the smaller the SMD SMD is not recommended when SDs differ considerably recognize that formula in text estimates SMD – exact formulas needed for actual research effect size for means and its calculation – divide by SD of control group Eta squared ???? tells you proportion of explained variance Chapter 10 Are all relationships causal? (NO) Confounding (third) variables affect the pattern of association between two other variables can affect size of association, direction of association, presence vs. absence of assoc. Experimental designs ???? researcher manipulates environment Survey designs ???? researcher only takes measurements Recognize random (methods of chance used ) vs. nonrandom assignment (any other method used) to groups Know that random assignment = randomization is the single characteristic that defines the true experiment (= randomized trial, randomized clinical trial) know that if study uses random assignment random assignment ... one need not be concerned about confounding variables thus (oversimplifying some), one may draw causal conclusions random assignment eliminates systematic bias pragmatically, it eliminates confounding variables (key point) technically, it randomly distributes confounding variables6 it does not create exactly equal groups groups differ due to luck of the random assignment process Designs with random assignment are much better for drawing causal conclusions than are ones without it. Among all quantitative designs, survey designs are probably regarded as the poorest for drawing causal conclusions. Chapter 11 “control for” confounding variable and “hold variable constant” = synonyms or very nearly so understand basic idea: if association disappears when one controls for confounding variable, the association is due to (caused by) that variable understand basic idea: if association weakens when one controls for confounding variable, the association is partly due to (caused by) that variable understand basic idea: if the size of association remains the same when the (potential) confounding variable is controlled for, the association is not due to the confounding variable (indeed, as the controlled for variable does not influence the association, it is not a confounding variable) understand that, within the subgroups, the controlled for variable becomes (essentially) a constant and, thus, cannot influence the strength of association between the variables in the initial relationship the within subgroup relationship is, in essence, the relationship in the absence of influence from the controlled for variable recognize the basic patterns when a third variable is controlled for (association persists, association weakens, association disappears, association differs in different subgroups) recognize direct and indirect effects recognize direct and indirect causal relationships recognize antecedent variable model recognize intervening variable model know that it is often very difficult to know which causal model is correct know that if antecedent variable model is correct, there is no causal effect whatsoever (see Figure 11.1) know that if intervening variable model is correct, there is no direct effect but there is an indirect effect (see Figure 11.2) (indirect causal relationship) recognize an interaction effect such as in Table 11.4 vs. the absence of (Table 11.5) – see Q23 Recognize that in a survey or other nonrandomized design, it is never possible to control for all confounding variables ???? ability to draw causal conclusions is always limited Chapter 12 Inferential statistics: tools for drawing conclusions about the population from which a sample is randomly selected not valid for samples that are not random (see exception in Chap. 16) characteristics of random sample selection by method of chance, equal chance of selection, independence of selection recognize a random (probability) sample vs. nonrandom (nonprobability) sample7 methods of chance used to draw random sample Independence of observations -- better described by example, so be able to recognize an example [in essence, that observations don=t share some unintended similarity; also the idea that no observation affects any other observation – see obvious examples in Q9] recognize that random sampling eliminates systematic sampling bias this is the great advantage of a random sample recognize that, due to luck of the draw (chance/random factors), characteristics of random samples almost always differ from those of the population from which selected recognize sample statistics vs. population parameters good estimators (inferential statistics) are efficient and unbiased statistics ???? Roman (English) letters; parameters ???? (often) Greek letters point estimates state a specific value interval estimates state a range of possible values know that the mean, standard deviation, variance, and proportion in a random sample provide the best estimates of their corresponding parameters in the population(with small samples, these estimates may not necessarily be good/close ones, but they are the best we can do) [standard deviation has negligible bias; others are unbiased] use point estimates to estimate population parameters – see Qs 25—28 be able to calculate sampling error – see Q31 Understand that sampling distributions are frequency distributions composed of the statistics of many random samples Know key points of central limit theorem Be able to describe the mean, SD, and shape of the sampling distribution of the mean; be able to do problems involving this – e.g. Q41 as sample size increases … the SD of the sampling distribution of the mean decreases sample statistics provide increasingly accurate estimates of population parameters sampling error / chance/ luck of the draw / randomness decrease Chapter 13 definition of confidence interval: probable range within which population parameter is located using formulas: calculations of 95% and 99% confidence intervals of the mean calculate estimate of standard deviation of sampling distribution (standard error of the mean) suppose that one extends a line two standard errors (= two standard deviations of sampling distribution) in both directions from study sample mean what % of such lines include population parameter: about 95% do ???? have 95% confidence for any given one Is researcher positive that confidence interval includes the population parameter – N0 Is researcher confident that CI includes population parameter – YES 95% confident if 95%CI; 99% confident if 99%CI given equal sample size (and other factors equal) which is wider 95% or 99% CI? -- 99%CI as sample size goes up (other factors staying same) the width of the CI ______ . decreases in the long run, what % of 95% CIs include the population parameter (95%) (99% for 99%CI) so you have 95% confidence that any given 95% CI does so (and 99% confidence B = greater confidence B that any given 99% CI does so)8 Be able to do confidence interval problems for proportions Points in section 13.5; in particular … margins of error = different way for expressing a confidence interval which is the more important in formulating confidence interval, the size of the sample or the size of the population from which the sample is selected? [size of sample] Nonresponse bias CIs used for many situations besides means and proportions Chapter 14 basic probability calculations -- 10 socks in drawer, 6 are black; probability of picking at random a pair of black socks = 6/10 = 0.6 also probability calculations for normal distribution (with aid of a figure) probability ranges from 0.00 (no chance) to 1.00 (certainty) Know that hypotheses pertain to the population from which sample (randomly) selected Know that null most often states the absence of association in the population Know that alternative most often states the presence of association in the population Know what the null and alternative most often state for different combinations of variables Know that the alternative “negates” the null Know that if reject null, (always) fail to reject (accept) the alternative Know that if fail to reject (accept) null, (always) reject the alternative Know that rejection of null and statistical significance go Ahand in hand@ (are the same thing) Know that acceptance of null and absence of statistical significance go Ahand in hand@ Know that rejection of null conveys that … study sample result provides strong evidence null is false study sample result is unlikely given a true null the null is likely false study sample result is unlikely to be due to chance alone one can be confident though not certain null is false. Know that failure to reject null conveys that … study sample result does not provide strong evidence null is false given a true null, study sample result is not a highly unlikely one chance alone is not a highly unlikely explanation for the study sample result Know that failure to reject null does not convey that … null is true, one may be confident null is true, or that null is likely true Know when to fail to reject (accept) the null and when to reject it Fail to reject when p > the selected statistical significance level (> alpha) Reject when p # the selected stat. sig. level (# alpha) Know that statistical significance test results (p, p value) convey ... given a true null, the probability of obtaining the study sample result or an even more extreme result the probability of obtaining the study sample result or an even more extreme result due to chance alone Stated more briefly, p conveys … given a true null, the probability of obtaining the study sample result the probability that the study sample result is due to chance alone Know that the just-mentioned probabilities are one and the same (the same thing)9 Is it harder to reject at the .01 level or the .05 level? (.01 level) reject at .01 level ???? 99% confident that null is false (that result not due to chance) reject at .05 level ???? 95% confident that null is false (that result not due to chance) Know that the null provides the simplest explanation for the study result asserts that sampling error alone is the explanation science will accept a more complex explanation --the alternative-- only when a simpler one does not suffice Be able to answer questions like Q40 and 46--49 Recognize nondirectional vs. directional hypothesis pairs and know when to use each Nondirectional pairs are more common Know which logical operators go with which hypothesis pair equal / not equal ???? nondirectional pair greater than / less than ???? directional pair In directional pair, alternative hypothesis states the direction of expectations Chapter 15 assumptions = conditions that must be met for a significance test to yield precisely accurate results tests can be robust to an assumption robust ???? violation of assumption affects probability given by test hardly at all ???? can conduct test even if assumption violated know that nondirectional hypothesis pairs go with two-tailed tests know that directional hypothesis pairs go with one-tailed tests know symbols for null H0 and alternative H1 and also statistical symbols in formulas be able to use the central limit theorem to describe the sampling distribution – its mean, SD, and shape -- given a true null understand rejection regions (size of and where located in different situations) know that two-tailed test (nondirectional hypothesis pair) has two rejection regions, one in each tail know that one-tailed test (directional pair) has only one rejection region this region is in direction of expectations = direction stated in the alternative know that (other things being equal and assuming that result is in expected direction) rejection of null is easier with a one-tailed test know that, assuming that result is in expected direction, one can calculate p for one-tailed test by dividing p for two-tailed test in half type l error = the error of rejecting a true null; equal to the selected significance level (= alpha) know that, given a true null, one expects to make a type I error 5% of time if .05 level is used 1% of time if .01 level is used type ll error = error of accepting a false null (see next chapter for more info) use of .01 level (rather than .05) decreases risk of type I error but increases that of type ll error know how to calculate z in the large sample test use the decision making rules given in the Chapter (rules will be provided on test) know that when chance alone is an unlikely (an insufficient) explanation, one may consider Areal@ explanations for the study result reject null ???? chance alone is an unlikely explanation ???? something real is at work realize that it is harder to reject the null at the .01 level than at the .05 level10 rejection regions are smaller and are located further from the value stated in the null Chapters 16 general idea of power = does researcher have a good opportunity to reject a false null definition of power = probability of rejecting a false null power of .80 is the standard for acceptable power the probability of type ll error is symbolized by Beta (β) Power = 1 – β (be able to use this formula); As sample size goes up (other things equal), the width of the sampling distribution decreases ???? chance / sampling error exerts less influence on study sample results Job of stat sig test = determine whether study sample result is due to chance or to something real As sample size goes up – due to reduced sampling error – job of test becomes easier As sample size goes up (other things being equal), power goes up ???? it becomes easier to reject a false null With large samples ???? even small associations in sample may be statistically significant in adoption study, N = 599, a correlation of -0.10 was statistically significant (p 252) With huge samples even trivial@/tiny associations may be statistically significant see academic achievement test example at end of section 16.4.2, p 252 With small samples ???? even large associations in sample may not be significant see out-of-home placement example on pp 250-252 Know factors other than sample size that affect power In particular, when a large difference / association rather than a small one is expected, power is enhanced – see Q14 Other factors that increase power: use of one-tailed rather than two-tailed test, use of .05 rather than .01 level, not having greatly reduced / restricted variability Control for extraneous variability pretests often increase power Be able to read power table on p 259 know that power goes up as … sample size goes up ???? as one goes down the columns size of association in population goes up ???? as one goes across the rows reality: significance tests are often conducted in the absence of random sampling know what p conveys = probability = probability that study result is due to chance alone in formal reports, best to report exact value of p recognize that fact of statistical significance = a low p -- does not convey … size of association, causality, generalizability, importance THE FOLLOWING COMMENT APPLIES TO ALL STATISTICAL TESTS and thus to Chapter 15 and Chapters 17 - 23: It is important to know which test to use in which situation (for instance, that an independent samples t test examines means from two independent samples, that a chi-square test of independence examines association between two categorical variables, etc. -- see info/table at end of this study guide)11 Chapter 17 N < 100 ???? for inferential statistical procedures that involve means … the t distribution rather than the normal distribution is the appropriate distribution N < 100: a key reason for using t distribution rather than normal … cannot be confident that the estimate of the standard deviation of the sampling distribution is accurate t distribution takes this inaccuracy into account and, thus, yields accurate results degrees of freedom: number of independent values after mathematical restrictions applied be able to calculate value of “missing” case in example such as Q4 t distribution is a family of distributions there is a different t distribution for each different degrees of freedom each one has a different shape for confidence intervals and single-sample t test: df = N - 1 (will be given) sample size very small -- say, N < 10 ???? t distribution and normal distribution have very different shapes : tails of t are more elongated and bigger/thicker As sample size increases: shape of the t distribution becomes increasingly similar to that of the normal distribution N > 100 ???? shape of t distribution nearly identical to that of normal The value of t within which, say, 95% of cases are located differs for different t distributions The changing shape of the t distribution is key to its accurate results Know how to carry out formulas for confidence interval and the one-sample t test Know when to use confidence intervals based on t and when to use the one-sample t test (rather than large sample test of the mean) = when N < 100 Needed decision rules for the one sample t test will be provided on the test do need to know how to use t distribution table (A.2) for confidence intervals and one-sample t test combination of very small sample size and very strong skew ???? confidence interval and one-sample t test can be inaccurate given Both directional (one-tailed test) and nondirectional hypothesis pairs (two-tailed test) are used for the one-sample test same pairs as for large sample test of the mean12 Chapter 18 know when to use the independent samples t test as opposed to the one-sample t test (when there are two independent samples) know that the sampling distribution of the difference between means is the key distribution for both independent and dependent samples t tests know the hypothesis pairs same for both of these tests ???? nondirectional null: means of two populations are equal both directional (one-tailed test) and nondirectional (two-tailed test) pairs used both tests assume normal distribution in population both tests are quite robust to this assumption very small sample size and very strong skew ???? results can be inaccurate two formulas for independent samples t test: equal variances formula and unequal variances formula with equal sample sizes, independent samples t test is robust to equality of variances assumption unequal sample sizes ???? independent samples t test not robust to equality of variances assump. use equal variances formula when equality of variances test fails to reject null of equal population variances when this test rejects the null ???? use the unequal variances formula df for independent samples t test: if equal variances: df = n1 + n2 - 2 (formula will be given) be able to calculate t (standard error of difference between means (denominator) will be given) be able to use t distribution table (A.2) (decision rules will be given) be able to distinguish between independent and dependent samples examples of dependent samples: pretest / posttest, partners in a relationship, etc. df for dependent samples test = N - 1, where N is number of pairs (formula will be given) know that positive correlation between the samples increases the power of the dependent samples t test know various symbols in formulas Chapter 19 One sample test of p: does sample proportion differ from a hypothesized/stated proportion? same decision rules as large sample test of X̅ directional (two-tailed test) or nondirectional (one-tailed test hypothesis pairs smaller of NP and N(1-P) should be ≥ 5 for sufficient accuracy: know how to compute be able to compute standard error of p: ���� = √(��(1 − ��))/�� (formula provided) be able to carry out test know symbols: e.g. p = proportion in sample, P = hypothesized proportion in population Binomial test tests same question as one-sample test of p whether sample proportion differs from a stated/hypothesized proportion good test when smaller of NP and N(1-P) < 5 except when P = .50: nondirectional pair ???? two-tailed probability not straightforward be able to use binomial test table > row ???? reject null if frequency in sample > number in table < row ???? reject null if frequency in sample < number in table13 One-variable chi-square test examines whether two or more proportions (percentages) observed in sample differ significantly from proportions stated in null based on chi-square distribution: key distribution for inferential procedures involving proportions Null: proportions in sample equal proportions stated in the null Given a true null, we expect the value of chi-square to be approximately equal to df chi-square is much larger than this ???? the null is likely false. Know (and be able to calculate) expected proportions (proportions in the null), expected frequencies (frequencies expected given a true null); to calculate: multiply expected proportion by the sample size : fe = pe × N, observed proportions (the actual proportions), and observed frequencies (the actual frequencies). Be able to carry out test Be able to use df formula to calculate df Reject null when value of obtained chi-square exceeds value in the chi-square table Even though hypothesis pair is always non-directional, chi-square test is a one-tailed test; rejection region is always in the upper tail Chapter 20 Chi-square test of independence: best known statistical test Examines whether association between two categorical variables is due to chance Know the two ways to present the null two variables are unassociated percentages are equal df = ( R - 1 ) ( C - 1) (be able to calculate; formula will be given) be able to calculate … observed column proportions (the proportion of column=s cases in the cell); observed frequencies (cell frequencies) expected column proportions (the proportion of cases expected given a true null simply the proportion in right margin or also: pe = (row margin total)/N); expected frequencies (formula will be given) Fe = [(row margin total)×(column margin total)] / N Also know how to calculate average frequency (divide N by number of cells) At most one chi-square calculation problem: will likely ask you to carry out just some section of the formula (formula will be given) Just as for one-variable chi-square test, the null is rejected when the obtained chi-square exceeds the value in the chi-square distribution table (table A.4) Hypothesis pair is always nondirectional; test is always one-tailed; rejection region is always in the upper tail. Chi-square test is a test of significance, not a measure of size of association14 Chapter 21 ANOVA examines differences between two or more sample means Null = all population means are equal Know the symbols used (Assuming 3 or more groups), the null is always nondirectional ANOVA is a one-tailed test; rejection region is in upper tail of F distribution MSW = estimate of population variance based on variance within groups (if equal Ns is equal to the average variance in each sample) MSB = estimate of population variance based on variance between groups If null is true, we expect MSW and MSW to have approximately equal values null is true ???? only sampling error causes these to differ If null is false (population means differ), we expect the MSB to be larger than the MSW null is false ???? both sampling error and real differences in population means affect MSB F = MSB/MSW (be able to calculate) null is true ???? expect F to equal about 1.00; null is false ???? expect F to be greater than 1.00 df for MSW = N - J ; df for the MSB = J - 1 (formula provided) be able to use the F distribution table (reject the null when obtained F is greater than the F in the table; know how to use table) Be able to calculate the MSB and MSW respectively from the SSW and SSW in ANOVA table (see Q29 and Q30; formulas provided) Problem with carrying out multiple independent t tests to see which differences significant … probability of a type l error for at least one test in a series of tests exceeds that for any single test Multiple comparisons: adjust significance level to account for the conducting of many comparisons /tests (Given a true null), when one carries out many tests, the probability that at least one test will reject the null exceeds this probability for any single test Many tests conducted ???? be skeptical of results (particularly when only selected results presented) Fishing: Author’s opinion: too much is bad, but some is OK; need balance Chapter 22 Test of r examines the probability that a correlation in study sample may be due to chance Nondirectional and directional hypothesis pairs Two-tailed (nondirectional hypothesis pair) and one-tailed (directional hypothesis pair) tests Nondirectional null: r = 0.00 Robust to normality assumption df = N - 2 (will be given) know how to use table of critical values of r (decision rules will be provided as needed) (In nondirectional pair, two-tailed test) … as sample size increases the absolute value of r necessary to reject decreases In real world research, significance reported at lowest achieved level APA recommends reporting actual (exact) probability rather than just significance level Confidence intervals often calculated for procedures other than proportions and means15 Parametric test ???? assumes a particular distributional shape for population(s) (usually normal) Nonparametric test ???? makes no assumption about distributional shape Parametric test ???? interval/ratio level dependent variable Nonparametric test ???? information actually used not at interval-ratio level Test of Spearman’s r correlation between rank orderings often used when sample size very small and very high degree of skew Mann-Whitney U test examines whether ranks differ significantly between two independent samples alternative to independent samples t test when sample size very small and skew very high Kruskal-Wallis test extends Mann-Whitney test to two or more groups alternative to ANOVA when sample size very small and skew very high Wilcoxon signed-ranks test: alternative to dependent samples t test Sign test = good alternative to dependent samples t test when data is categorical McNemar’s test -- examines significance of a change of proportions in two dependent samples Data transformation apply a mathematical manipulation to each score often an alternative to a nonparametric test square root and log transformations often applied to positively skewed dependent variable with goal of making that variable have a distributional shape much closer to normal ???? can carry out parametric test of transformed data Single-case designs baseline (A) vs. intervention (B) phase researcher treats baseline data as one group and intervention data as the other nominal-level dependent variable ???? chi-square test of independence often good choice in general, offers researcher some ability to draw causal conclusions though less so than in randomized design serial dependency (in essence) similarity between observations close together in time in categorical data: revealed by long streaks, clusters in numeric data: serpentine pattern reveals possible serial dependency if present: tests in this text may be inaccurate (specialized tests required) Qualitative researchers often hunt for qualitative themes they sometimes perform statistical analyses on these themes ???? become quantitative Reasoning with data significant result ???? chance alone an unlikely explanation absence of random assignment and significant result ???? result may be due to confounding variable ???? poor causality significant result and random assignment significance rules out chance (though not with certainty) and … random assignment rules out confounding variable basically: researcher can be confident that intervention is cause of result researcher does not know what particular aspect of intervention is causal agent result not significant can’t rule out chance as an unlikely explanation (chance alone is sufficient)16 ???? don’t even begin to think about issues related to causality when sample size is very small … poor power (rather than actual absence of relationship in population) often responsible for failure to reject null Note: not responsible for any material in Chapters 23 and 24 -- but do see last page Chapter 23 Multiple regression B = nonstandardized coefficient ???? interpret using raw scores (original measurement metric) as predictor (independent variable) increases by 1 (1.00), what is the predicted change in the dependent variable, with all other predictor variables in equation held constant (controlled for)? β (beta) = standardized coefficient B equivalent to r except that now we have several predictor variables in the equation, and the others are controlled for B as predictor goes up by 1 (1.00) standard deviation, what is the predicted change in the dependent variable in standard deviation units, with all other predictor variables in equation held constant? β ???? does convey size of association; B does not do so dummy variable: a dichotomous (binary) variable with its categories coded as 0 and 1 the (nearly) ideal situation in multiple regression: all predictors have strong association to dependent variable and … all predictors have small/weak correlations with one another ???? explains (nearly) maximum variance, (nearly) maximum prediction, and (nearly) highest power (given sample size, etc.) with respect to individual predictors not a good situation: very high correlation between predictors variance that each predictor explains overlaps with that explained by other each additional variable explains little new variance reduced power to detect effect of any given variable may still want to include a highly correlated variable if it is a potential confounding variable multicollinearity: extremely high correlation between (combination of) variables regression may “crash” power to detect contribution of individual predictors is greatly reduced logistic regression: excellent approach when dependent variable is dichotomous (binary) equation predicting odds rather than log of odds easier to interpret (exponentiated) coefficients convey, controlling for other predictors in equation, the predicted change in the odds as predictor increases by 1.00 two-way ANOVA (factorial ANOVA) factors = the independent variables and … levels = categories of the factors e.g. 3 x 4 ANOVA = two-way ANOVA; first factor has 3 levels; 2nd has 417 big advantage: can test for interaction effect between the two factors drop the interaction term if it is not significant Procedures related to factorial ANOVA … ANCOVA ???? adds interval/ratio-level predictors (covariates) MANOVA ????examines several predictors in a single analysis MANCOVA ???? adds covariates to MANOVA Repeated measures design ???? examines several administrations of same dependent variable in a single design analysis Selected advanced procedures structural equation models: causal models often with multiple measures of variables multililevel modeling: examines contributions of variables at multiple levels of analysis survival analysis (event history analysis): preferred when dependent variable is time to occurrence factor analysis: reduces multiple pieces of information to a lesser number Chapter 24 Generalizability: expected similarity between study result and result that would be obtained in a different context statistical generalizations generalizations made with inferential statistical procedures may only be made to population --if any-- from which study sample randomly selected this, of course, is great limitation of statistical generalization given adequate sample size, can be made with considerable confidence this is advantage/ strong point nonstatistical generalizations generalizations based on nonstatistical logic guided predominantly by degree of similarity must be made very cautiously; in essence require a cautious leap of faith beyond the data this, of course, is great disadvantage: in general, a broad diverse study sample enhances generalizability similar results in different sample subgroups ???? good generalizability across these groups replication carry out study multiple times in varied settings key study features are strategically varied similar results as features varied ???? increased confidence in generalizability differing results as features varied ???? results may not generalize across feature highly important in assessing generalizability Importance: importance of study result in real world a relationship can be statistically significant and unimportant a relationship can be not statistically significant, but, nevertheless, important Importance ultimately requires nonstatistical (judgment) rather than statistical logic18 Balanced data interpretation model (1) statistical significance -- whether the relationship is likely to be due to chance; (2) size (strength) of association: the degree to which the values of one variable vary, differ, or change according to the values of the other; (3) causality -- whether one variable causes the other; (4) generalizability -- the degree to which a similar relationship or study result will be found in a different context or setting; and (5) importance -- the real-world implications of study results. Key to model: keep parts distinct do not assume that because a result is statistically significant it is also large, causal, generalizable and/or important. do not assume that because a result is not statistically significant, it is also small, not generalizable, and/or unimportant* *note: if result not significant, one doesn’t conclude that it causal, as it may be due to chance alone Author’s recommendation: given the factors reasonably equal importance = balance NEXT PAGE PRESENTS GUIDELINE FOR WHICH TEST IN WHICH SITUATION19 Which statistical test in which situation? Notes: multivariate tests not listed here. Listed tests are key tests; however it is possible that questions about a test not listed here could be asked.
When Test is Used
One-sample t test
Determine whether difference between sample mean and some stated value is statistically significant
Independent samples t test
Determine whether difference between means in two independent samples is statistically significant
Dependent samples t test
Determine whether difference between means in two dependent samples is statistically significant
One-sample test of a proportion
Determine whether difference between sample proportion (percentage) and some stated proportion (percentage) is statistically significant
Determine whether difference between sample proportion and some stated proportion (percentage) is statistically significant
One-variable chi-square test
Determine whether two or more proportions (percentages) in study sample differ to a statistically significant degree from hypothesized proportions (percentages)
Chi-square test of independence
Determine whether association observed between two categorical variables in study sample is statistically significant (and also: whether (column) proportions (percentages) in study sample differ to a statistically significant degree)
Analysis of variance (ANOVA)
Determine whether difference(s) between means in two or more independent samples is/are statistically significant
Test of Pearson=s r
Determine whether correlation in sample differs to a statistically significant degree from 0.00
what conclusions can and cannot be drawn for variables at different levels?
what are the key characteristics of each level?
more questions on the four levels of measurement than on anything else in Chapter 1 what are these levels?
We also discuss several other topics like Define toxicity.
If you want to learn more check out cotton is one of the oldest crops grown in texas.
We also discuss several other topics like What refers to the quantitative description of the structure of populations?
We also discuss several other topics like What is a coelum?
Don't forget about the age old question of How did the Greek view the afterlife?
If you want to learn more check out ole miss social work