T-tests Part 2 PS1006 Lecture 3 Sam Cromie

From Repeated t to Unrelated t NOMENCLATURE within, repeated, paired vs. between, unrelated, independent

Generic form of a statistic Data – Hypothesis Error What you got – what you expected (null) The unreliability of your data

Repeated measures t test 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

SPSS repeated t output

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

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

Changing to between groups design 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

= = = =

Equation elements = mean of group 1 = mean of group 2 = the standard deviation of a sampling distribution based on the difference between the mean of two samples = the variance of group 1 = the variance of group 2 n1= the number of participants in group 1 n2= the number of participants in group 2

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 such that and are replaced by

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

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

SPSS output Note SPSS uses the pooled variance formula

Degrees of freedom Each group has 9 participants New result 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

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

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

Conditions of use Formula Value interested in Population value Denominator Pop mean and SD known - interested in score of ind Score of individual Population mean Population standard deviation Pop mean and SD known - interested in mean of sample Mean of sample Standard error of sampling distribution of mean Pop mean but SD unknown - interested in mean of sample SE of sampling distribution of mean Interested in difference between 2 repeated measures Mean difference between two repeated measures zero SE of sampling distribution of mean difference scores Interested in difference between 2 independent Gs Difference between means of 2 independent Gs SE of differences between means