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

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PSY 340 Statistics for the Social Sciences 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

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PSY 340 Statistics for the Social Sciences Statistical analysis follows design The related-samples t- test can be used when: –1 sample –Two scores per subject

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PSY 340 Statistics for the Social Sciences participantsPre-testPost-test Test 2-levels, All of the participants are in both levels of the IV levels Sometimes called “repeated measures” design Within-Groups Factor Memory patients before getting the treatment Memory patients after getting the treatment

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PSY 340 Statistics for the Social Sciences Statistical analysis follows design The related-samples t- test can be used when: –1 sample –Two scores per subject –2 samples –Scores are related - OR -

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PSY 340 Statistics for the Social Sciences Matching groups Group AGroup 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 Blue tall 23yrs Green average 22yrs Brown tall 22yrs Color Height Age matched Red Short 21yrs matched Blue tall 23yrs matched Green average 22yrs matched Brown tall 22yrs

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PSY 340 Statistics for the Social Sciences Testing Hypotheses –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 Hypothesis testing: a five step program Very similar to one sample t-test from earlier

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PSY 340 Statistics for the Social Sciences Performing your statistical test What are we doing when we test the hypotheses? –Computing a test statistic: Generic test Could be difference between a sample and a population, or between different samples Based on standard error or an estimate of the standard error Compares the differences between groups of related observations to the difference expected by the null hypothesis Because the groups of observations come from same individuals or matched individuals, the variability is typically reduced

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PSY 340 Statistics for the Social Sciences Performing your statistical test Diff. Expected by chance Estimated standard error of the differences Test statistic What are all of these “D’s” referring to? Mean of the differences Number of difference scores Difference scores –For each person, subtract one score from the other –Carry out hypothesis testing with the difference scores H 0 Population of difference scores has a mean = 0

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PSY 340 Statistics for the Social Sciences Comparing t-tests One-sample t Independent-samples t Observed (sample) means Related samples t Difference between Observed (sample) means

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PSY 340 Statistics for the Social Sciences Comparing t-tests One-sample t Independent-samples t Related samples t Hypothesized population means from the Null hypothesis Hypothesized difference between Population means from the Null hypothesis

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PSY 340 Statistics for the Social Sciences Comparing t-tests One-sample t Independent-samples t Related samples t Hypothesized difference between Population means from the Null hypothesis H0:H0: Memory performance by the treatment group is equal to memory performance by the no treatment group. So:

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PSY 340 Statistics for the Social Sciences Comparing t-tests One-sample t Independent-samples t Related samples t Hypothesized difference between Population means from the Null hypothesis H0:H0: Memory performance by the treatment group is equal to memory performance by the no treatment group. So: Hypothesized 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. = 0 Is equal to The major difference between the two tests comes from how the estimated standard errors are computed.

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PSY 340 Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Difference scores Person Pre-test Post-test (Pre-test) - (Post-test) H0:H0: There is no difference between pre-test and post- test HA:HA: There is a difference between pre-test and post- test μ D = 0 μ D ≠ 0

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PSY 340 Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Difference scores = 5.5 Person Pre-test Post-test (Pre-test) - (Post-test)

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PSY 340 Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person Pre-test Post-test Difference scores D = 5.5

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PSY 340 Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person Difference scores Pre-test Post-test D = = = = = = SS D D - D(D - D) 2

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PSY 340 Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person Pre-test Post-test D = = SS D (D - D) D - D Difference scores

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PSY 340 Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person Pre-test Post-test D = = SS D (D - D) = s D D - D Difference scores

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PSY 340 Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person Pre-test Post-test D = = SS D (D - D) = s D 1.45 = s D ? Think back to the null hypotheses D - D Difference scores

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PSY 340 Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person Pre-test Post-test D = = SS D (D - D) = s D 1.45 = s D H 0 : Memory performance at the post-test are equal to memory performance at the pre-test D - D Difference scores

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PSY 340 Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person Pre-test Post-test D = = SS D (D - D) = s D 1.45 = s D This is our t obs D - D Difference scores

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PSY 340 Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person Pre-test Post-test D = = SS D (D - D) = s D 1.45 = s D t obs α= 0.05 Two-tailed t crit D - D Difference scores

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PSY 340 Statistics for the Social Sciences = t crit - Reject H 0 Performing your statistical test What are all of these “D’s” referring to? Person Pre-test Post-test D = = SS D (D - D) = s D 1.45 = s D t obs α= 0.05 Two-tailed t crit D - D Difference scores t obs =3.8

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PSY 340 Statistics for the Social Sciences Performing your statistical test What are all of these “D’s” referring to? Person Pre-test Post-test D = = SS D (D - D) = s D 1.45 = s D t obs α= 0.05 Two-tailed t crit T obs > t crit so we reject the H D - D Difference scores

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PSY 340 Statistics for the Social Sciences Statistical Tests Summary DesignStatistical test (Estimated) Standard error One sample, σ known One sample, σ unknown Two related samples, σ unknown Two independent samples, σ unknown

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PSY 340 Statistics for the Social Sciences Effect Sizes & Power for t Test for Dependent Means Remember we don’t know these 7-11

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PSY 340 Statistics for the Social Sciences Approximate Sample Size Needed for 80% Power (.05 significance level) Using Power and effect sizes to determine how many participants you need 7-12

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PSY 340 Statistics for the Social Sciences Using SPSS: Related samples t Entering the data –Different groups of observations go into separate columns e.g., pre-test in one column, post-test in a separate column Person Pre-test Post-test 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.)

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