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Introduction To Statistics In Psychology

by: Celia O'Hara

Introduction To Statistics In Psychology PSY 20100

Marketplace > Purdue University > Psychlogy > PSY 20100 > Introduction To Statistics In Psychology
Celia O'Hara
GPA 3.93

Gregory Francis

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Gregory Francis
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This 6 page Class Notes was uploaded by Celia O'Hara on Saturday September 19, 2015. The Class Notes belongs to PSY 20100 at Purdue University taught by Gregory Francis in Fall. Since its upload, it has received 12 views. For similar materials see /class/207865/psy-20100-purdue-university in Psychlogy at Purdue University.


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Date Created: 09/19/15
Introduction to Statistics in PSY 201 Professor Greg Francis Lecture 35 ANalysis Of VAriance The last top is ASSUMPTIONS to use ANOVA validly the data must me t 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 3 The variances of the distributions in thepopulatiohs 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 1 State the hypothesis Ho M1M2 MKiHoiM 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 normality have small e ects on Type I error rates violations of homogeneity of variance have a big e ect if the population sizes are di erent means that ANOVA is robust as long as the sample 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 means is statistically signi cant it does not tell us if it is important one measure that helps us determine importance is the strength of association to measures association between independent and dependent variables similar to correlation coe icient of determination 7 SIGNIFICANCE What percentage of the variation in the data is accounted for by t e independent variable changes in groups Mg 7 335 7 K 71MSW SST MSW in our example problem last time We plug in the numbers 2 7 408074 7 227991 1079852 27991 03178 the di erent lecture styles account for approximately 32 of the variance suggests that the results are not really that strong a t tests if We have only two groups Ho Mi M2 Ha M1 M2 We can use either ANOVA or the tetests discussed previously they give identical results 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 t2 F ti 7 Fm so using either technique produces the same results reject or not EXAMPLE A sociologist wants to detemine whether sorority donnitory women date more o iie men who liv e in and 12 women who live in dormitories tes they each onth The ioiiowmg nits have during the ensuing m are the res reject 1 TEST 1 TEST ANOVA so standard error is H0 M M2 Ha M1 M2 test with a 005 twoetailed Ho M1 7 M2 0 Ha i M1 M2 0 We have equal numbers of subjects so We do not need to test for homogeneity of variance df n1n272 1212e2 22 so from Table CB We nd tw i2074 from data We calculate the pooled estimate of population variance 52 5570 10 t 7 1038 2074 t 0963 lt m which is not in the region of rejection so do not reject H0 no evidence for a di erence in number of dates We have 1 df in the numerator and 22 df in the denominator so from ble 05 FM 430 We can calculate SSB 600 SSW 122500 SST 128500 check 12850 SST SSBSSW 600122500 ANOVA 335 600 MS 7 7 600 B K 7 1 1 33 122500 MSW 7W 7 5568 N 7 M55 600 F MSW 5568 139078 since F 1078 lt 430 Fm We do not reject H0 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 331 330 SSRES where I SST is the total sum of square I 331 is the variation among indir vidua s note school F 1078 m 1077 103592 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 i 39 39 ua s i 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 216710316 302 where X EzXZk k n is the average for the kth observation across all subjects 330 deviation of observation mean from overall mean similar to 355 in the normal We need a term that correspond to SSW We can directly calculate the total sum o squares SST 2ltsz 7W 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 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 33R across observations 0 M SR F 55 K 71X 71 if H0 is true MSRES estimates the variance of the gt if H is true population distribution HO 39 M1 M2 MK 0 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 K 71n 71 330 39 M30 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 otherwise it overestimates it EXAMPLE 1 HYPOTHESIS A school principal traces reading rehension 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 Third Grade Fbuxth Grade th Grade 1 2 8 a 2 45 10 50 2 26 40 3 31 43 4 38 49 5 25 31 6 24 31 7 32 38 8 30 36 77 2340 3000 2ka 36143 k 1 Ho i M1 M2 M3 Ha MZ Mkfor some i and 76 test with a 005 2 CRITICAL VALUE 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 lormulcs m the book we nd SST gch e 7 where T ls the sum 01311 scores so 2 SST 36143 790210 36143 7 342 77 SST 18 66 and T3 T2 5539 1 W where T is the sum ol obsewations lor the 2th ndiyidual so SS34844734277567 Note this cannot be negatiyel 3 TEST STATISTIC T T2 33 i if 0 N 2 2 2 330 23 40 30 0 37 30 7342 77 8 330 354 86 7 342 77 12 09 so any remaining yariation is residual SSE SST e ssr e 330 33Re51866756771209090 again these cannot be negatiyel 3 TEST STATISTIC now calculate O 1209 MS 7 7 605 0 K71 2 and 33p 7 090 Mspgs 7 006 K 71n71 7 14 and get the F statistic M50 5 10083 60 F 7 7 Mngs 006 4 INTERPRET since F 10083 gt 374 Fm We reject Ho there is evidence that the reading scores for these subjects are dl erent across the years SUMMARY TABLE meof Sumo u reesl arian variation squares lreedom estimate Fratio 777T 100 83 Q E e a 006 a 3 ga fgo wgarre 2 o005 g 4 T ereforeHols reiected ASSUMPTIONS ANOVA for repeated measures depends on four assumptions 1 The sample was randomly selected for a population 2 The dependent variable eg reading scores is normally distributed in the population 3 The population yariances for the test occasions are equal homogeneity of variance 4 The population correlation coef cients between pairs of t ual est occasion scores deyiation from 2 tends to not cause serious problems deyiations from 3 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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