Two-sample t-tests

+14 M = 14 Flood M = -36.3 -27 -41

Independent-samples t-test Often interested in whether two groups have same mean Experimental vs. control conditions Comparing learning procedures, with vs. without drug, lesions, etc. Men vs. women, depressed vs. not Comparison of two separate populations Population A, sample A of size nA, mean MA estimates mA Population B, sample B of size nB, mean MB estimates mB mA = mB? Example: maze times Rats without hippocampus: Sample A = [37, 31, 27, 46, 33] With hippocampus: Sample B = [43, 26, 35, 31, 28] MA = 34.8, MB = 32.6 Is difference reliable? mA > mB? Null hypothesis: mA = mB No assumptions of what each is (e.g., mA = 10, mB= 10) Alternative Hypothesis: mA ≠ mB

Finding a Test Statistic Goal: Define a test statistic for deciding mA = mB vs. mA ≠ mB Constraints (apply to all hypothesis testing): Must be function of data (both samples) Sampling distribution must be fully determined by H0 Can only assume mA = mB Can’t depend on mA or mB separately, or on s Alternative hypothesis should predict extreme values Statistic should measure deviation from mA = mB so that if mA ≠ mB, we’ll be able to reject H0 Answer (preview): Based on MA – MB (just like M – m0 for one-sample t-test) . (MA – MB) has Normal distribution Standard error has (modified) chi-square distribution Ratio has t distribution

Likelihood Function for MA – MB Central Limit Theorem Distribution of MA – MB Subtract the means: E(MA – MB) = E(MA) – E(MB) = m – m = 0 Add the variances: . Just divide by standard error? Same problem as before: We don’t know s Need to estimate from data

Estimating s Already know best estimator for one sample Could just use one sample or the other sA or sB Works, but not best use of the data Combining sA and sB Both come from averages of (X – M)2 Average them all together: Degrees of freedom (nA – 1) + (nB – 1) = nA + nB – 2

Independent-Samples t Statistic Difference between sample means Typical difference expected by chance Variance of MA – MB Estimate of s2 Variance from MA Variance from MB Sum of squared deviations Degrees of freedom Mean Square; estimates s2

Steps of Independent Samples t-test State clearly the two hypotheses Determine null and alternative hypotheses H0: mA = mB H1: mA ≠ mB Compute the test statistic t from the data . Difference between sample means, divided by standard error Determine likelihood function for test statistic according to H0 t distribution with nA + nB – 2 degrees of freedom Choose alpha level Find critical value 7a. t beyond tcrit: Reject null hypothesis, mA ≠ mB 7b. t within tcrit: Retain null hypothesis, mA = mB

Example Rats without hippocampus: Sample A = [37, 31, 27, 46, 33] With hippocampus: Sample B = [43, 26, 35, 31, 28] MA = 34.8, MB = 32.6, MA – MB = 2.2 df = nA + nB – 2 = 5 + 5 – 2 = 8 t8 tcrit = 1.86 X X-MA (X-MA)2 37 2.2 4.84 31 -3.8 14.44 27 -7.8 60.84 46 11.2 125.44 33 -1.8 3.24 SA(X-MA)2 = 208.80 X X-MB (X-MB)2 43 10.4 108.16 26 -6.6 43.56 35 2.4 5.76 31 -1.6 2.56 28 -4.6 21.16 SB(X-MB)2 = 181.20

Mean Squares Average of squared deviations Used for estimating variance Population Population variance, s2 Sample Sample variance, s2 Estimates s2 Two samples Also estimates s2

Degrees of Freedom Applies to any sum-of-squares type formula Tells how many numbers are really being added n = 2: only one number In general: one number determined by the rest Every statistic in formula that’s based on X removes 1 df M, MA, MB Algebraically rewriting formula in terms of only X results in fewer summands I will always tell you the rule for df for each formula To get Mean Square, divide sum of squares by df Sampling distribution of a statistic depends on its degrees of freedom c2, t, F X X – M (X – M)2 3 -2 4 7 2

Independent vs. Paired Samples Independent-samples t-test assumes no relation between Sample A and Sample B Unrelated subjects, randomly assigned Necessary for standard error of (MA – MB) to be correct Sometimes samples are paired Each score in Sample A goes with a score in Sample B Before vs. after, husband vs. wife, matched controls Paired-samples t-test

Paired-samples t-test Data are pairs of scores, (XA, XB) Form two samples, XA and XB Samples are not independent Same null hypothesis as with independent samples mA = mB Equivalent to mean(XA – XB) = 0 Approach Compute difference scores, Xdiff = XA – XB One-sample t-test on difference scores, with m0 = 0

Example Breath holding underwater vs. on land 8 subjects Water: XA = [54, 98, 67, 143, 82, 91, 129, 112] Land: XB = [52, 94, 69, 139, 79, 86, 130, 110] Difference: Xdiff = [2, 4, -2, 4, 3, 5, -1, 2] Critical value > qt(.025,7,lower.tail=FALSE) [1] 2.364624 Reliably longer underwater Mean: Mean Square: Standard Error: Test Statistic:

Comparison of t-tests Samples Data t Standard Error Mean Square df One X n - 1 2-Indep. XA, XB nA + nB – 2 2-Paired Xdiff = XA - XB