Chapter 12: Comparing Independent Means 12/3/2018

In Chapter 12: 12.1 Paired and Independent Samples 12/3/2018 In Chapter 12: 12.1 Paired and Independent Samples 12.2 Exploratory and Descriptive Statistics 12.3 Inference About the Mean Difference 12.4 Equal Variance t Procedure (Optional) 12.5 Conditions for Inference 12.6 Sample Size and Power 12/3/2018 Basic Biostat

Types of Samples Single sample. One group; no concurrent control group Paired samples. Two samples; data points uniquely matched Two independent samples. Two samples, separate (unrelated) groups. 12/3/2018

What Type of Sample? Measure vitamin content in loaves of bread and see if the average meets national standards Compare vitamin content of loaves immediately after baking versus content in same loaves 3 days later Compare vitamin content of bread immediately after baking versus loaves that have been on shelf for 3 days Answers: 1 = single sample 2 = paired samples 3 = independent samples 12/3/2018

Experimental vs. Observational Groups Independent samples can Experimental –an intervention or treatment is assigned as part of the study protocol Non-experimental (observational) – groups defined by a innate characteristics or self-selected exposure “Two Groups” by Pieter Bruegel the Elder (c. 1525 – 1569) 12/3/2018

Do means from these populations differ? If so, by how much? Illustrative Data* * Data set WCGS.sav (p. 49) Type A personality men (n = 20) 233, 291, 312, 250, 246, 197, 268, 224, 239, 239, 254, 276, 234, 181, 248, 252, 202, 218, 212, 325 Type B personality men (n = 20) 344, 185, 263, 246, 224, 212, 188, 250, 148, 169, 226, 175, 242, 252, 153, 183, 137, 202, 194, 213 Do means from these populations differ? If so, by how much? 12/3/2018

Notation Statistics (sample) Group 1 n1 s1 Group 2 n2 s2 Parameters (population) Group 1 N1 µ1 σ1 Group 2 N2 µ2 σ2 12/3/2018

Illustrative Data Cholesterol levels (mg / dL) Group n mean std dev 1 20 245.05 36.64 2 210.30 48.34 Type A men in the sample have higher average cholesterol by 35 mg/dL 12/3/2018

Standard Error To address this question, calculate the standard error of the mean difference: 12/3/2018

Degrees of Freedom Two ways to estimate degrees of freedom: dfWelch [complex formula on p. 244 of text] dfconserv. = the smaller of (n1 – 1) or (n2 – 1) For the illustrative data: dfWelch = 35.4 (via SPSS) dfWelch = 35.4 (via SPSS) dfconserv. = smaller of (n1–1) or (n2 – 1) = 20 – 1 = 19 dfconserv. = smaller of (n1–1) or (n2 – 1) = 20 – 1 = 19 12/3/2018

(point estimate) ± (t)(SE) (1 – α)100% CI for µ1–µ2 Note: (point estimate) ± (t)(SE) margin of error 12/3/2018

Comparison of CI Formulas Type of sample Point estimate df for t* SE Single n – 1 Paired Independent smaller of n1−1 or n2−1 12/3/2018

Example 12/3/2018

Interpretation The CI interval aims for µ1 − µ2 with (1– α)100% confidence 12/3/2018

Hypothesis Test Test claim of “no difference in populations” Note: widely different sample means can arise just by chance Null hypothesis: H0: μ1 – μ2 = 0 (equivalently H0: μ1 = μ2) Alternative hypothesis Ha: μ1 – μ2 ≠ 0 (two-sided) OR Ha: μ1 – μ2 > 0 (“right-sided”) OR Ha: μ1 – μ2 < 0 (“left-sided”) 12/3/2018

Test Statistic dfWelch= 35.4 (via SPSS) dfconserv. = 19 12/3/2018

P-value via Table C tstat = 2.56 with 19 df One-tailed P between .01 and .005 Two-tailed P between .02 and .01 (i.e., less than .02) .01 < P < .02 provides good evidence against H0  observed difference is statistically significant 12/3/2018

SPSS Response variable (chol) in one column Explanatory variable (group) in a different column 12/3/2018

SPSS Output Equal Variance Not Assumed Preferred method (§12.3) Equal variance t procedure (§12.4) Equal Variance Not Assumed Preferred method (§12.3) 12/3/2018

Summary of independent t test H0: μ1 –μ2 = 0 C. P-value from Table C or computer (Interpret in usual fashion) 12/3/2018

Hypothesis Test with the CI H0: μ1 – μ2 = 0 can be tested at α-level of significance with the (1 – α)100% CI Example: 95% CI for μ1 – μ2 = (6.4 to 63.1)  excludes μ1 – μ2 = 0  Significant difference at α = .05 12/3/2018

Hypothesis Test with the CI 99% CI for μ1 – μ2 is (-2.2 to 71.7), which includes μ1 – μ2 = 0  Not Significant at α = .01 12/3/2018