Report

T-tests Part 2 PS1006 Lecture 3 Sam Cromie 1 From Repeated t to Unrelated t NOMENCLATURE within, repeated, paired vs. between, unrelated, independent 2 Generic form of a statistic Data – Hypothesis Error What you got – what you expected (null) The unreliability of your data 3 Repeated measures t test Before Mean St. Dev. 21 24 21 26 32 27 21 25 18 23.84 4.20 D t sD After 15 15 17 20 17 20 8 19 10 15.67 4.24 D sD n Diff. 6 9 4 6 15 7 13 6 8 8.22 3.60 • PTSD symptoms measured before and after supportive counseling • Difference scores are used for the calculation • t calculates the likelihood of achieving these scores (using the concept of a sampling distribution), given there is there is no difference between before and after scores • Since there should be no difference we assume (pop diff score) to be 0 8.22 3.6 9 8.22 6.85 1.2 4 SPSS repeated t output Paired Samples St at ist ics Pair 1 BEFORE AFTER Mean 23.89 15.67 N Std. Deviation 4.20 4.24 9 9 Std. Error Mean 1.40 1.41 Paired Samples Correlations N Pair 1 BEFORE & AFTER 9 Correlation .637 Sig. .065 Paired Samples Test Paired Differences Pair 1 BEFORE - AFTER Mean 8.22 Std. Deviation 3.60 Std. Error Mean 1.20 95% Confidence Interval of the Difference Lower Upper 5.46 10.99 t 6.856 df 8 Sig. (2-tailed) .000 5 Reporting the result • Supportive counselling resulted in a decrease (M= 8.22, SD=3.6) in the number of PTSD symptoms reported. A repeated measures t test showed these differences to be significant; t(8)=6.86, p<.001, two-tailed. – shorthand t(8)=6.86, p<.001, two-tailed – In exam - conclusion = supportive counselling reduced the number of PTSD symptoms • Reporting p Options are: > .05, <.05, <.01, <.001 Never state that p =.000 or that p is < .000 6 Independent groups t test • Used to analyse a between subjects design – Also referred to as a between subjects t test • Should realise our therapy trial could have been designed using two different groups rather than a repeated measures design – One group received therapy the other did not • There are no comparable scores within each group therefore groups as a whole have to be compared 7 Changing to between groups design Mean St. Dev. No Therapy Group Therapy Group 21 24 21 26 32 27 21 25 18 15 15 17 20 17 20 8 19 10 23.84 4.20 15.67 4.24 • Same data presented as different groups • No comparable scores within each group - groups as a whole have to be compared • Test differences between sample means • Need a sampling distribution of differences between group means 8 D t sD X t 1 X 2 ( 1 2 ) sX 1 X 2 8.22 3.6 9 8.22 4.2 2 4.242 9 9 = D sD n X1 X 2 = 2 1 2 2 s s n1 n2 = 8.22 6.85 1.2 = 8.22 4.13 1.99 9 Equation elements • X 1 = mean of group 1 • X 2 = mean of group 2 •sX X = the standard deviation of a sampling distribution based on the difference between the mean of two samples 2 s • 1 = the variance of group 1 2 s • 2 = the variance of group 2 • n1= the number of participants in group 1 • n2= the number of participants in group 2 1 2 10 Allowing for Gs of different sizes • A sample variance should be weighted according to the number within the sample • Formula below calculates the pooled variance 2 2 2 such that s1 and s2 are replaced by s p t X1 X 2 s 2p n1 s 2p n1 X1 X 2 1 1 s n1 n2 2 p where 2 2 ( n 1 ) s n 1 s 1 2 2 s 2p 1 n1 n2 2 11 Inputting data into SPSS • Basic rule - each participant occupies a single row – Repeated measures design: • each participant = 2 columns, 1 for before and 1 for after therapy – Between groups design: • all the scores go into 1 column since each participant only produces one score 12 • With between groups each participant must also be identified in terms of the group they come from • A second column is designated the grouping variable (sometimes referred to as dummy variable) - identifying which group the participant was in 13 SPSS output Group St atist ics GROUP BEFORE 1 2 N Mean 23.89 15.67 9 9 Std. Deviation 4.20 4.24 Std. Error Mean 1.40 1.41 Independent Samples Test Levene's Test for Equal ity of Var iances F BEFORE Equal vari ances assumed Equal vari ances not assumed .000 Sig . .983 t- test for Equal ity of Means t 4.133 4.133 df 16 15.998 Sig . ( 2-tai led) .001 Mean Difference 8.22 Std. Error Differ ence 1.99 .001 8.22 1.99 95% Confi dence Inter val of the Di fference Lower Upper 4.01 12.44 4.01 12.44 • Note SPSS uses the pooled variance formula 14 Degrees of freedom • Each group has 9 participants – df for each group = n - 1 = 9 - 1 = 8 – Since there are 2 groups • df = n1 - 1 + n2 - 1 = n1 + n2 - 2 • = 9 + 9 - 2 = 16 df • New result t(16) = 4.133, p<.01, two-tailed – Value of t is smaller - independent groups design is less powerful and will always produce a smaller t result given the same data 15 Conditions of use For all parametric statistics, the data must fulfil three criteria with varying stringency – The data must be of interval quality – Both populations are sampled from populations with equal variances • Homogeneity of variance – Both groups are sampled from normal populations • Assumption of normality 16 Nonparametric equivalents • When the data produced do not conform to the requirements of parametric data, then there are nonparametric equivalents • Repeated measures t test – Wilcoxon’s Matched-Pairs Signed-Ranks Test • Unrelated groups t test – Mann-Whitney (U) Test 17 Conditions of use Pop mean and SD known - interested in score of ind Formula X z X Value interested in Score of individual Population value Denominator Population mean Population standard deviation Population mean Standard error of sampling distribution of mean Population mean SE of sampling distribution of mean Pop mean and SD known - interested in mean of sample z Pop mean but SD unknown interested in mean of sample X t sX Mean of sample Interested in difference between 2 repeated measures D t sD Mean difference between two repeated measures zero SE of sampling distribution of mean difference scores Interested in difference between 2 independent Gs X t Difference between means of 2 independent Gs zero SE of differences 18 between means 1 X X 2 ( 1 2 ) sX 1 X 2 Mean of sample