Statistics for the Social Sciences Psychology 340 Spring 2010 Using t-tests (related samples)

t Test for Dependent Means Unknown population mean and variance More realistic, these designs are commonly found in research, Based on having two related groups of observations Two situations One sample, two scores for each person Repeated measures design Two samples, but individuals in the samples are related Related samples, Dependent samples, matched samples

Statistical analysis follows design 1 sample Two scores per subject The related-samples t-test can be used when:

Within-Groups Factor Sometimes called “repeated measures” design 2-levels, All of the participants are in both levels of the IV levels participants Pre-test Post-test Test Memory patients before getting the treatment Memory patients after getting the treatment

Statistical analysis follows design The related-samples t-test can be used when: 2 samples Scores are related 1 sample Two scores per subject - OR -

Matching groups Group A Group B Matched groups Trying to create equivalent groups Also trying to reduce some of the overall variability Eliminating variability from the variables that you matched people on Red Short 21yrs matched Red Short 21yrs matched Blue tall 23yrs Blue tall 23yrs matched Green average 22yrs Green average 22yrs Color Height Age matched Brown tall 22yrs Brown tall 22yrs

Testing Hypotheses Hypothesis testing: a five step program Step 1: State your hypotheses Step 2: Set your decision criteria Step 3: Collect your data Step 4: Compute your test statistics Compute your estimated standard error Compute your t-statistic Compute your degrees of freedom Step 5: Make a decision about your null hypothesis Very similar to one sample t-test from earlier

Performing your statistical test What are we doing when we test the hypotheses? Computing a test statistic: Generic test Compares the differences between groups of related observations to the difference expected by the null hypothesis Could be difference between a sample and a population, or between different samples Based on standard error or an estimate of the standard error Because the groups of observations come from same individuals or matched individuals, the variability is typically reduced

Performing your statistical test Difference scores For each person, subtract one score from the other Carry out hypothesis testing with the difference scores H0 Population of difference scores has a mean = 0 What are all of these “D’s” referring to? Mean of the differences Test statistic Diff. Expected by chance Estimated standard error of the differences Number of difference scores

Comparing t-tests Independent-samples t One-sample t Related samples t Observed (sample) means Difference between Observed (sample) means

Comparing t-tests Independent-samples t One-sample t Related samples t Hypothesized population means from the Null hypothesis Hypothesized difference between Population means from the Null hypothesis

Comparing t-tests Independent-samples t One-sample t Related samples t Hypothesized difference between Population means from the Null hypothesis H0: Memory performance by the treatment group is equal to memory performance by the no treatment group. So:

Comparing t-tests Independent-samples t One-sample t Related samples t Hypothesized difference between Population means from the Null hypothesis The numerator’s of both the independent samples and related samples t-tests are numerically identical. I’ve used notational differences to illustrate that one is based on two sets of observations (from the two samples), while the other is based on difference scores. Hypothesized population means from the Null hypothesis = 0 = 0 Is equal to H0: Memory performance by the treatment group is equal to memory performance by the no treatment group. The major difference between the two tests comes from how the estimated standard errors are computed. So:

Performing your statistical test (Pre-test) - (Post-test) What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 1 2 3 4 45 55 40 60 43 49 35 51 2 H0: There is no difference between pre-test and post-test 6 5 9 μD = 0 22 HA: There is a difference between pre-test and post-test μD ≠ 0

Performing your statistical test (Pre-test) - (Post-test) What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 1 2 3 4 45 55 40 60 43 49 35 51 2 6 5 9 22 = 5.5

Performing your statistical test What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 1 45 43 2 2 55 49 6 3 40 35 5 4 60 51 9 22 D = 5.5

Performing your statistical test What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 45 55 40 60 43 49 35 51 D - D (D - D)2 1 2 - 5.5 = -3.5 12.25 2 6 0.5 - 5.5 = 0.25 3 5 -0.5 - 5.5 = 0.25 4 9 3.5 - 5.5 = 12.25 22 25 = SSD D = 5.5

Performing your statistical test What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 45 55 40 60 43 49 35 51 D - D (D - D)2 1 2 -3.5 12.25 2 6 0.5 0.25 3 5 -0.5 0.25 4 9 3.5 12.25 22 25 = SSD D = 5.5

Performing your statistical test What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 45 55 40 60 43 49 35 51 D - D (D - D)2 1 2 -3.5 12.25 2 6 0.5 0.25 3 5 -0.5 0.25 4 9 3.5 12.25 22 25 = SSD D = 5.5 2.9 = sD

Performing your statistical test What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 45 55 40 60 43 49 35 51 D - D (D - D)2 ? 1 2 -3.5 12.25 2 6 0.5 0.25 3 5 -0.5 0.25 4 9 3.5 12.25 Think back to the null hypotheses 22 25 = SSD D = 5.5 2.9 = sD 1.45 = sD

Performing your statistical test What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 45 55 40 60 43 49 35 51 D - D (D - D)2 1 2 -3.5 12.25 2 6 0.5 0.25 H0: Memory performance at the post-test are equal to memory performance at the pre-test. 3 5 -0.5 0.25 4 9 3.5 12.25 22 25 = SSD D = 5.5 2.9 = sD 1.45 = sD

Performing your statistical test What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 45 55 40 60 43 49 35 51 D - D (D - D)2 1 2 -3.5 12.25 2 6 0.5 0.25 This is our tobs 3 5 -0.5 0.25 4 9 3.5 12.25 22 25 = SSD D = 5.5 2.9 = sD 1.45 = sD

Performing your statistical test What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 45 55 40 60 43 49 35 51 D - D (D - D)2 1 2 -3.5 12.25 2 6 0.5 0.25 tobs 3 5 -0.5 0.25 tcrit α= 0.05 Two-tailed 4 9 3.5 12.25 22 25 = SSD D = 5.5 2.9 = sD 1.45 = sD

Performing your statistical test What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 45 55 40 60 43 49 35 51 D - D (D - D)2 1 2 -3.5 12.25 2 6 0.5 0.25 tobs 3 5 -0.5 0.25 tcrit α= 0.05 Two-tailed 4 9 3.5 12.25 +3.18 = tcrit 22 25 = SSD tobs=3.8 D = 5.5 2.9 = sD - Reject H0 1.45 = sD

Performing your statistical test What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test 45 55 40 60 43 49 35 51 D - D (D - D)2 1 2 -3.5 12.25 2 6 0.5 0.25 tobs 3 5 -0.5 0.25 tcrit α= 0.05 Two-tailed 4 9 3.5 12.25 22 25 = SSD Tobs > tcrit so we reject the H0 D = 5.5 2.9 = sD 1.45 = sD

Statistical Tests Summary Design Statistical test (Estimated) Standard error One sample, σ known One sample, σ unknown Two related samples, σ unknown Two independent samples, σ unknown

Effect Sizes & Power for t Test for Dependent Means 7-11 Remember we don’t know these

Approximate Sample Size Needed for 80% Power (.05 significance level) Using Power and effect sizes to determine how many participants you need 7-12

Using SPSS: Related samples t Person Pre-test Post-test Entering the data Different groups of observations go into separate columns e.g., pre-test in one column, post-test in a separate column 1 45 43 2 55 49 3 40 35 4 60 51 Performing the analysis Analyze -> Compare means -> paired samples t-test Identify which columns are the related samples of obs Reading the output Means of the different groups, the mean difference, the computed-t, degrees of freedom, p-value (Sig.)