Parametric & Nonparametric Models for Within-Groups Comparisons• overview• WG Designs & WG RH:/RQ:• McNemar’s test • Cochran’s tests• WG t-test & ANOVA• Wilcoxin’s test• Friedman’s F-testStatistics We Will ConsiderParametric NonparametricDV Categorical Interval/ND Ordinal/~NDunivariate stats mode, #cats mean, std median, IQRunivariate tests gofX21-grp t-test 1-grp Mdn testassociation X2Pearson’s r Spearman’s r2 bg X2 t- / F-test M-W K-W Mdnk bg X2 F-test K-W Mdn2wgMcNem Crn’s t- / F-test Wil’s Fried’skwg Crn’s F-test Fried’sM-W -- Mann-Whitney U-Test Wil’s -- Wilcoxin’s Test Fried’s -- Friedman’s F-testK-W -- Kruskal-Wallis TestMdn -- Median Test McNem -- McNemar’s X2Crn’s – Cochran’s Test Repeated measures designs…There are two major kinds of these designs:1) same cases measured on the same variable at different times or under different conditions• pre-test vs. post-test scores of clients receiving therapy• performance scores under feedback vs. no feedback conds• % who “pass” before versus after remedial training 2) same cases measured at one time under one condition, using different (yet comparable) measures• comparing math and reading scores (both T-scores, withmean=50 and std=10)• number of “omissions” (words left out) and “intrusions” (words that shouldn’t have been included) in a word recall task• % who “pass” using two different testsRepeated measures designs…There is really a third related kind of design:3) non-independent groups of cases measured on the same variable at different times or under different conditions• matched-groups designs• snow-ball sampling over timeStatistically speaking, groups-comparisons analyses divide into 2 kinds”• independent groups designs Æ Between Groups designs• dependent groups designs Æ within-groups & Matched-groups designs For all dependent groups designs, the non-independence of the groups allows the separation of variance due to “differences among people”from variance due to “unknown causes” (error or residual variance)For repeated measures designs (especially of the first 2 kinds),there are two different types of research hypotheses or questions that might be posed…1) Do the measures have different means (dif resp dist for qual DVs)• are post-test scores higher than pre-test scores?• is performance better with feedback than without it?• are reading scores higher than math scores?• are there more omissions than intrusions?2) Are the measures associated?• are the folks with the highest pre-test scores also the ones with the highest post-test scores?• is performance with feedback predictable based onperformance without feedback?• are math scores and reading scores correlated?• do participants who make more omissions also tend tomake more intrusions?So, taken together there are four “kinds of” repeated measures analyses. Each is jointly determined by the type of design and the type of research hypothesis/question. Like this…Type of Hypothesis/QuestionType of Design mean difference associationDifferent times or pre-test < post-test pre-test & post-test situationsDifferent measures math < spelling math & spellingBut… All the examples so far have used quantitative variables. Qualitative variables could be used with each type of repeated measures design (dif times vs. dif measures)Consider the difference between the following examples of repeated measures designs using a qualitative (binary) response oroutcome variable• The same % of students will be identified as needing remedial instruction at the beginning and end of the semester (dif times).•The same students will be identified as needing remedial instruction at the end of the semester as at the beginning (dif times)• The same % of folks will be identified as needing remedial instruction based on teacher evaluations as based on a standardized test (dif measures)•The same folks will be identified as needing remedial instruction based on teacher evaluations as based on a standardized test (dif measures)So, we have to expand our thinking to include 8 situations... So, for repeated measures designs, here are the analytic “situations” and the statistic to use for eachType of Question/HypothesisQuant Vars Qual VarsType of Design mean dif assoc % dif^ pattern^*Different times wg t/F-test Pearson’s r Cochrans McNemar’s X²or situationsDifferent wg t/F-test Pearson’s r Cochran’s McNemar’s X²measures ^ Cochran’s and McNemar’s are for use only with binary variables* McNemar’s looks at patterns of classification disagreementsMcNemar’s testOf all these tests, McNemar’s has the most specific application…• are two qualitative variable related -- Pearson’s X2• do groups have differences on a qual variable -- Pearson’s X2• does a group change % on a binary variable -- Cochran’s• is the the relationship between the variables revealed by an asymmetrical pattern of “disagreements” –McNemar’se.g., more folks are classified as “pass” by the computer test but “fail” by the paper testthan are classified as “fail” by the computer test but “pass” by the paper test.computer testpaper test pass failpass 40 4fail 12 32 Statistical Tests for WG Designs w/ qualitative variablescomputer testpaper test pass failpass 40 4fail 12 32 Cond #1Cond #2 value 1 value 2value 1 a bvalue 2 c d (b – c)2X2=(b + c)(4 – 12)2X2 =(4 + 12)= 4Compare the obtained X2with X2 1, .05= 3.84. We would reject H0: and conclude that there is a relationship between what performance on the paper test and performance on the computer test & that more uniquely fail the paper test than uniquely fail the computer test.McNemar’s always has df=1Cochran’s Q-test – can be applied to 2 or k-groupsThe simplest “qualitative variable” situation is when the variable is binary. Then “changes in response distribution” becomes the much simple “changes in %”.Begin the computation of Q by arranging the data with each case on a separate row.