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# 44 Outline for PSY 20100 with Professor Francis at Purdue

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Date Created: 02/06/15

Introduction to Statistics in Psychology PSY 201 Professor Greg Francis Lecture 35 ANalysis Of VAriance The last topic ASSUMPTIONS to use ANOVA validly the data must meet some restrictions 1 The observations are random and in dependent samples from the popular tions 2 The distributions of the populationsfrom which samples are selected are nor ma 3 The variances of the distributions in thepopulations are equal Homogene ity of variance How do we check these assumptions 1 Sample properly 2 test 3 Bartlett s test ANOVA TESTING 4 STEPS IState the hypothesis Ho M1M2 MKlHoiM M for some i7j 2 Set the criterion FltK7LN7Kgt 3 Compute the test statistic F M SB MSW 4 Interpret results ASSUMPTIONS it turns out that violations of norrnality have srnall e ects on Type I error rates violations of hornogeneity of variance have a big e ect if the population sizes are di erent rneans that ANOVA is robust as long as the sarnple sizes are the same across populations Note though the restrictions to apply ANOVA are more stringent than for tatests no restrictions on population distributions SIGNIFICANCE ANOVA tells us if an e ect di erence of rneans is statistically signi cant it does not tell us if it is irnportant one rneasure that helps us determine irnportance is the strength of association p12 rneasures association between independent and dependent variables sirnilar to correlation coe icient of deterrnination 7 2 SIGNIFICANCE What percentage of the variation in the data is accounted for by the independent variable changes in groups Mg 335 e K 71MSW SST MSW in our exarnple problern last tirne We plug in the nurnbers 2 7 408074 7 227991 1079852 27991 03178 the di erent lecture styles account for approxirnately 32 of the variance suggests that the results are not really that strong a t tests if We have only two groups Ho Ml M2 Ha M1 M2 We can use either ANOVA or the t tests it turns out that the F distribution for K 7 17N 7 K 17 N 7 2 degrees of freedom is simply the t distribution for N 7 2 df squared EXAMPLE A sociologlst wants to determrne whether sorority or dormrtory women date more olten lle randomly samples 12 women who live m sororities and 12 women who live m dormrtorres and determmes the number ol dates they each have during the ensuing month The lollowmg are the results tetests dlscussed preVlously t2 F Sommy Doxmmw 2 Women X1 Women X2 they give identical results tw Fm 8 g 5 7 so using either technique produces 639 3 the same results reject or not 42 reject 7 8 9 7 10 5 5 8 3 6 7 3 5 5 7131 T2 69 716 750 272 m 750 t TEST t TEST ANOVA test With a 005 twoetalled so standard erlror ls1 HO M M2 HOiMl M20 57132 EVE Haim tz EmmiL27 7 770963 I we have equal numbers of 5X17X2 39 We haye 1 df ln the numerator and and 22 df in the denominator so from subjects so We do not need to test 7 7 T M C 5 for homogeneity of variance W a e df n1n272 1212e2 22 10 5331in FM 430 so from Table CB We nd 1038 lt 2074 tau we can Calculate tw i2074 from data We calculate the pooled estimate of population variance 52 5570 which is not in the region of rejection so do not reject H0 no evidence for a di erence in number of dates SSE 600 SSW 122500 SST 128500 check 12850 SST SSBSSW 600122500 ANOVA 335 500 7 500 K 7 r r SSW 122500 MS W N 7 K 22 M35 500 MSW 5568 139078 since F 1078 lt 430 Fm We do not reject H0 M55 5568 REPEATED MEASURES one Way ANOVA deals with independent samples We Want to consider one situation with a dependency suppose we repeat our measurements from the same individuals at di erent times eg tracking health patterns across years grades throughout SUM OF SQUARES scores for an individual are dependent scores for di erent individuals are independent SST 531 330 55R where I SST is the total sum of square I 331 is the variation among indir note school Viduals F 1078 m 1077 10382 t2 I 330 is the variation among test Fm 430 430 20742 1 occasions I SSRES is any other type of varir ation INDIVIDUALS OBSERVATIONS RESIDUAL the combined variation among individuals is 351 21K X 7 302 where EkX k X Z Z K is the average for the ith individual across all observations SSI deviation of individual means from overall mean does not correspond to SSW or 355 in the normal ANOVA the combined variation across observations is 350 215710315 302 where EzXZk n is the average for the kth observation across all subjects Xk SSO deviation of observation mean from overall mean similar to 355 in the normal ANOVA We need a term that correspond to SSW We can directly calculate the total sum of squares SST 22sz 732 1 if there is variation beyond 331 and 350 We can calculate it as SSRES SST 7 331 7 330 this is similar to SSW factors out variation due to individuals and variation due to observations VARIANCE ESTIMATES SSRES gt 0 due to random sampling choice of individuals VARIANCE ESTIMATES 330 can vary due to random sampling or due to di erences F RATIO as before We compare these estimates with the F statistic 35R across observations MSO MSRES MS K 71n 1 if H0 is true RES estimates the variance of the f H t population distribution HO 39 M1 M2 MK 1 O ls me I then there are no di erences across F N 10 the degr es Ofl eedlom 350C13ted observations so all variation must if H0 is not true With this esmmate ls be due to random sampling So F gt 1 O K71n71 MS 7 330 39 O i K 71 look up Fm for K71 Kelnrl estimates the variance of the degrees of freedom POPHIatiOH dismbution if H0 is everything else is the same as true before otherwise it overestimates it EXAMPLE 1 HYPOTHESIS 2 CRITICAL VALUE A school principal traces reading comprehension scores on a standardized test for a random sample of dyslexic students across three years The data are given below Complete the ANOVA using a 005 Student Third Grade Fburth Grade Flith Grade 72 1 2 8 3 2 4 5 10 50 2 2 6 4 0 5 1 1170 3 3 1 4 3 5 0 12 40 4 3 8 4 9 5 7 14 40 5 2 5 3 1 4 4 10 00 6 2 4 3 1 3 9 9 40 7 3 2 3 8 4 3 11 30 8 3 0 3 6 4 4 11 00 T4 23 40 30 00 37 30 90 70 2ka 36143 C 2 Ho iM1 M2 M3 Ha MZ Mkfor some i and 76 test with a 005 for the numerator observation sum of squares We have dfK713712 for the denominator residual sum of squares We have df K71n71 371x871 14 so from Table 05 We nd the F critical value to be Fm 374 3 TEST STATISTIC using iormulcs m the book we nd 3 TEST STATISTIC 3 TEST STATISTIC 33 TI T7 now calculate T2 0 Z i T SS 12 09 7 Z 7 7 ssTeggxa N 2 k H N 2 M30 0 7505 where T is the sum oi all scores 33 7 23 40 30 0 37 30 7 K 7 1 2 O 7 7 342 77 8 8 and so 330354867342771209 35 090 SST36143790702 36143734277 M5385 K 71521571 14 03906 24 so any remaining variation is residual SST 18 as and SSR 5SSTTSSITSSO and get the Fstatlstic T T2 SSR531866756771209090 M5 505 SS E W FMsiom10083 where T is the sum oi obseryauons ior the 2th again these cannot be negatiyel R 39 indiyiducl so ss348 447342 7567 Note this cannot be negatiyel rr rt r 4 INTERPRET SUMMARY TABLE ASSUMPTIONS since F 10083 gt 374 Fm We reject Ho there is evidence that the reading scores for these subjects are dl ererrt across the years Source oi Sum oi Degrees oi Variance variation squares ireedom estimate Frctio Occasions 1209 2 605 10083 lndiyiducls 5 s7 7 7 Residual 090 14 006 Total 1866 23 ith a 0 05 M 4 Thereiore H0 is rejected ANOVA for repeated measures depends on iour assumptions The sample was randomly selected for a population to The dependent variable eg reading scores is normally distributed in the population A The population yariances ior the test occasions are equal homogeneity of variance p The population correlation coef cients between pairs of test occasion scores are equal deviation from 2 tends to not cause serious problems deviations from a and 4 can be compensated for sometimes an CONCLUSIONS restrictions on use of oneeway ANOVA strength of association ANOVA repeated measures partitioning the variance accounts for dependence arnong scores for sarne subjects NEXT TIME nish up details of class Study guide for nal exam How does it feel to not be a schmuck

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