Lecture 8 PY 427 Statistics 1 Fall 2006 Kin Ching Kong, Ph.D Chicago School of Professional Psychology Lecture 8 Kin Ching Kong, Ph.D

Agenda The t Test for Two Independent Samples Hypothesis Testing The independent-measures or between-subjects design The hypotheses for an independent-measures t test The formula for an independent-measures t test Pooled Variance Estimated Standard Error The Degree of Freedom Hypothesis Testing Directional Tests Assumptions for the independent-measures t test Measuring Effect Size

Two Research Designs Most research involves comparing two or more sets of data. Two general research designs used to obtain two or more sets of data: Independent-measures or between-subjects design: each set of data come from a completely separate sample of individuals. In other words, a separate sample is used for each treatment condition or each level of the independent variable. Figure 10.1 of your book Repeated-measures or within-subjects design: two or more sets of data come from the same sample of individuals. In other words, the same sample is repeatedly measured across treatment conditions or levels of the independent variable. So far we talked about inferential statistics using one sample to make inferences about an unknown population.

The hypotheses for an independent-measures t test The independent-measures t test: Evaluate the mean difference between two populations (or between two treatment conditions) The Null Hypothesis: H0: m1 – m2 = 0 (there is no difference between the population means) The Alternative Hypothesis: H1: m1 – m2 = 0 (there is a mean difference)

The formula for an independent-measures t test The basic structure of the t Statistic: t = sample statistic – hypothesized population parameter estimated standard error t = data – hypothesis error The single-sample t t = M – m sM The independent-measures t t = (M1 – M2) – (m1 – m2) s(M1 – M2)

The Standard Error In general, standard error measure how accurately the sample statistic represents the population parameter. In other words, how much difference to expect between statistic and parameter by chance alone. sstatistic For the independent-measures t, the standard error measures the amount of error that is expected when you use a sample mean difference (M1 – M2) to represent a population mean difference (m1 – m2). s(M1 – M2) For a one sample t test, the statistic is M, so the standard error is sM

The Standard Error for independent-measures t Two sources of error: M1 approximates m1 with some error M2 approximates m2 with some error Standard error for each sample: sM = For two samples, the conceptual formula: s(M1 – M2) =

Pooled Variance Problems with the conceptual formula for s(M1 – M2) When n1 = n2, the formula is biased. The variance (s2) from the larger sample is a more accurate estimate of s2 and therefore should be given more weight (law of large numbers). The Pooled Variance s2p = SS1 + SS2 df1 + df2 Ans to 2: df = 5, t = + 2.571

Estimated Standard Error for independent-measures t An unbiased measure of the standard error: s(M1 – M2) = The pooled variance is used instead of the individual sample variances.

Interpreting the estimated standard error in independent-measures t The meaning of the estimated standard error: Average discrepancy between a sample statistic (M1 – M2) and the corresponding population parameter (m1 – m2) expect by chance. Estimated std error = average ((M1 – M2) - (m1 – m2)) expected by chance When H0 is true, m1 – m2 = 0: Estimated std error = average (M1 – M2) expected by chance Then: t = actual difference between M1 and M2 average difference (expected by chance) between M1 and M2

The Final Formula and df The independent-measures t t = (M1 – M2) – (m1 – m2) s(M1 – M2) s(M1 – M2) = s2p = SS1 + SS2 df1 + df2 The degree of freedom for the independent-measures t: df = df for the first sample + df for the second sample df = df1 + df2 = (n1 -1) + (n2 – 1)

Single Sample t vs Independent-Measures t Same conceptual structure for both t formulas: t = data – hypothesis error Same basic elements in both t formulas, but the independent-measures t doubles each aspect of the single-sample t: Table 10.1 of your book

Hypothesis Testing (the experiment) Research question: Does using mental images affect memory? Experiment: Participants: Two groups of 10 participants (2 separate samples) Stimuli: a list of 40 pairs of nouns (e.g. dog/bicycle, lamp/piano) Treatment (IV): Group 2: given the list for 5 minutes and told to memorize the noun pairs Group 1: same list for 5 minutes and told to memorize, but also told to form a mental image for each pair of nouns (e.g. dog riding a bicycle). Dependent Variable: Each group is given a memory test in which they are given the first word and told to recall the second word of each pair Number of words correctly recalled was recorded for each participant (DV)

Hypothesis Testing (the data) Group 1 (images) Group 2 (no images) 19 32 23 12 20 30 22 16 24 27 15 14 31 25 18 n1 = 10 n2 = 10 M1 = 26 M2 = 18 SS = 200 SS = 160

Hypothesis Testing (Steps 1 & 2) Step 1: State the hypotheses (& choose a alpha level) H0: m1 – m2 = 0 (no difference, imagery has no effect) H1: m1 – m2 = 0 (there is a difference, imagery has an effect) a = .05 Step 2: Define Critical Region df = df1 + df2 = (n1 - 1) + (n2 – 1) =9 + 9 = 18 For a = .05, the critical region consist of the extreme 5% of the distribution beyond t = +2.101, i.e. tcritical = +2.101 Fig. 10.2 of your book

Hypothesis Testing (Step 3) Step 3: Compute test statistic t = (M1 – M2) – (m1 – m2) s(M1 – M2) s(M1 – M2) = s2p = SS1 + SS2 df1 + df2 s2p = SS1 + SS2 = 200 + 160 = 360 = 20 df1 + df2 9+9 18 s(M1 – M2) = = = 2

Hypothesis Testing (Step 3 continue & Step 4) Step 3: Compute test statistic t = (M1 – M2) – (m1 – m2) s(M1 – M2) = (26 – 18) – 0 2 = 8/2 = 4.00 Step 4: Make a decision Since the obtained value (t = 4.00) is in the critical region (i.e. greater than +2.101), we reject H0 and concluded that using mental images produced a significant difference in memory performance. Another way to think about this: the obtained difference between the group is 4 times greater than would be expected by chance (the standard error). This result is very unlikely if the null hypothesis is true.

Effect Size for the independent-measures t Cohen’s d = mean difference standard deviation Cohen’s d = M1 – M2 d = 26 – 18 = 8/4.47 = 1.79 r2: percentage of variance explained by the treatment. r2 = t 2 = 42 = 16/34 = 0.47 t 2 + df 42 + 18 So 47% of the variance in the data are explained by the treatment effect.

Assumptions Three assumptions underlying the independent-measures t: The observations within each sample must be independent. The two populations from which the samples are selected must be normally distributed. The two populations from which the samples are selected must have equal variance (homogeneity of variance). Test for the homogeneity of variance If the largest variance is no more than 3 to 4 times the smallest variance, then assume homogeneity of variance F-max = s2 (largest)/s2(smallest) Use Table B.3 to find the critical value: k = number of separate samples df = n -1 (F-max test is used when all n’s are equal)