1. Sociology 601 Class 8: September 24, 2009 6.6: Small-sample inference for a proportion 7.1: Large sample comparisons for two independent sample means. 7.2: Difference between two large sample proportions. 1

2. 7.1 Large sample comparisons for two independent means So far, we have been making estimates and inferences about a single sample statistic Now, we will begin making estimates and inferences for two sample statistics at once. –many real-life problems involve such comparisons –two-group problems often serve as a starting point for more involved statistics, as we shall see in this class. 2

3. Independent and dependent samples Two independent random samples: –Two subsamples, each with a mean score for some other variable –example: Comparisons of work hours by race or sex –example: Comparison of earnings by marital status Two dependent random samples : –Two observations are being compared for each “unit” in the sample –example: before-and-after measurements of the same person at two time points –example: earnings before and after marriage –husband-wife differences 3

4. Comparison of two large-sample means for independent groups Hypothesis testing as we have done it so far: Test statistic: z = (Y bar -  o ) / (s /SQRT(n)) What can we do when we make inferences about a difference between population means (  2 -  1 )? –Treat one sample mean as if it were  o ? –(NO: too much type I error) –Calculate a confidence interval for each sample mean and see if they overlap? –(NO: too much type II error) 4

5. Figuring out a test statistic for a comparison of two means Is Y 2 –Y 1 an appropriate way to evaluate  2 -  1 ? Answer: Yes. We can appropriately define (  2 -  1 ) as a parameter of interest and estimate it in an unbiased way with (Y 2 – Y 1 ) just as we would estimate  with Y. This line of argument may seem trivial, but it becomes important when we work with variance and standard deviations. 5

6. Figuring out a standard error for a comparison of two means Comparing standard errors: A&F 213: formula without derivation Is s 2 Ybar2 - s 2 Ybar1 an appropriate way to estimate  2 (Ybar2-Ybar1) ? –No! –  2 (Ybar2-Ybar1) =  2 (Ybar2) - 2  (Ybar2,Ybar1) +  2 (Ybar1) –Where 2  (Ybar2,Ybar1) reflects how much the observations for the two groups are dependent. –For independent groups, 2  (Ybar2,Ybar1) = 0, so  2 (Ybar2-Ybar1) =  2 (Ybar2) +  2 (Ybar1) 6

7. Step 1: Significance test for  2 -  1 The parameter of interest is  2 -  1 Assumptions: –the sample is drawn from a random sample of some sort, –the parameter of interest is a variable with an interval scale, –the sample size is large enough that the sampling distribution of Y bar2 – Y bar1 is approximately normal. –The two samples are drawn independently 7

8. Step 2: Significance test for  2 -  1 The null hypothesis will be that there is no difference between the population means. This means that any difference we observe is due to random chance. H o:  2 -  1 = 0 –(We can specify an alpha level now if we want) Q: Would it matter if we used H o:  1 -  2 = 0 ? H o:  1 =  2 ? 8

9. Step 3: Significance test for  2 -  1 The test statistic has a standard form: –z = (estimate of parameter – H o value of parameter) standard error of parameter Q: If the null hypothesis is that the means are the same, why do we estimate two different standard deviations? 9

10. Step 4: Significance test for  2 -  1 P-value of calculated z: Table A Stata: display 2 * (1 – normal(z) ) Stata: testi (no data, just parameters) Stata: ttest (if data file in memory) 10

11. Step 5: Significance test for  2 -  1 Step 5: Conclusion. Compare the p-value from step 4 to the alpha level in step 1. If p < α, reject H 0 If p ≥ α, do not reject H 0 State a conclusion about the statistical significance of the test. Briefly discuss the substantive importance of your findings. 11

12. Significance test for  2 -  1 : Example Do women spend more time on housework than men? Data from the 1988 National Survey of Families and Households: –sexsample sizemean hourss.d –men425218.112.9 –women676432.618.2 The parameter of interest is  2 -  1 12

13. Significance test for  2 -  1 : Example 1.Assumptions: random sample, interval-scale variable, sample size large enough that the sampling distribution of  2 -  1 is approximately normal, independent groups 2.Hypothesis: H o :  2 -  1 = 0 3.Test statistic: z = ((32.6 – 18.1) – 0) / SQRT((12.9) 2 /4252 + (18.2) 2 /6764) = 48.8 4.p-value: p<.001 5.conclusion: a.reject H 0 : these sample differences are very unlikely to occur if men and women do the same number of hours of housework. b.furthermore, the observed difference of 14.5 hours per week is a substantively important difference in the amount of housework. 13

14. Confidence interval for  2 -  1 : housework example with 99% interval: c.i…. = (32.6 – 18.1) +/- 2.58*( √((12.9) 2 /4252 + (18.2) 2 /6764)) = 14.5 +/- 2.58*.30 = 14.5 +/-.8, or (13.7,15.3) By this analysis, the 99% confidence interval for the difference in housework is 13.7 to 15.3 hours. 14

15. Stata: Large sample significance test for  2 -  1 Immediate (no data, just parameters) –ttesti 4252 18.1 12.9 6764 32.6 18.2, unequal Q: why ttesti with large samples? For the immediate command, you need the following: –sample size for group 1 (n = 4252) –mean for group 1 –standard deviation for group 1 –sample size for group 2 –mean for group 2 –standard deviation for group 2 –instructions to not assume equal variance (, unequal) 15

16. Stata: Large sample significance test for  2 -  1, an example. ttesti 4252 18.1 12.9 6764 32.6 18.2, unequal Two-sample t test with unequal variances ------------------------------------------------------------------------------ | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] ---------+-------------------------------------------------------------------- x | 4252 18.1.1978304 12.9 17.71215 18.48785 y | 6764 32.6.221294 18.2 32.16619 33.03381 ---------+-------------------------------------------------------------------- combined | 11016 27.00323.1697512 17.8166 26.67049 27.33597 ---------+-------------------------------------------------------------------- diff | -14.5.2968297 -15.08184 -13.91816 ------------------------------------------------------------------------------ Satterthwaite's degrees of freedom: 10858.6 Ho: mean(x) - mean(y) = diff = 0 Ha: diff 0 t = -48.8496 t = -48.8496 t = -48.8496 P |t| = 0.0000 P > t = 1.0000 16

17. Large sample significance test for  2 -  1 : command for a data set (#1). ttest YEARSJOB, by(nonstandard) unequal Two-sample t test with unequal variances ------------------------------------------------------------------------------ Group | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] ---------+-------------------------------------------------------------------- 0 | 980 9.430612.2788544 8.729523 8.883391 9.977833 1 | 379 7.907652.3880947 7.555398 7.144557 8.670747 ---------+-------------------------------------------------------------------- combined | 1359 9.005887.2290413 8.443521 8.556573 9.4552 ---------+-------------------------------------------------------------------- diff | 1.522961.4778884.5848756 2.461045 ------------------------------------------------------------------------------ diff = mean(0) - mean(1) t = 3.1869 Ho: diff = 0 Satterthwaite's degrees of freedom = 787.963 Ha: diff 0 Pr(T |t|) = 0.0015 Pr(T > t) = 0.0007 17

18. Large sample significance test for  2 -  1 : command for a data set (#2). ttest conrinc if wrkstat==1, by(wrkslf) unequal Two-sample t test with unequal variances ------------------------------------------------------------------------------ Group | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] ---------+-------------------------------------------------------------------- self-emp | 190 48514.62 2406.263 33168.05 43768.03 53261.2 someone | 1263 34417.11 636.9954 22638 33167.43 35666.8 ---------+-------------------------------------------------------------------- combined | 1453 36260.56 648.5844 24722.9 34988.3 37532.82 ---------+-------------------------------------------------------------------- diff | 14097.5 2489.15 9191.402 19003.6 ------------------------------------------------------------------------------ diff = mean(self-emp) - mean(someone) t = 5.6636 Ho: diff = 0 Satterthwaite's degrees of freedom = 216.259 Ha: diff 0 Pr(T |t|) = 0.0000 Pr(T > t) = 0.0000 18

19. 7.2: Comparisons of two independent population proportions In 1982 and 1994, respondents in the General Social Survey were asked: “Do you agree or disagree with this statement? ‘Women should take care of running their homes and leave running the country up to men.’” –YearAgreeDisagreeTotal –1982122223345 –199426816321900 –Total 39018552245 Do a formal test to decide whether opinions differed in the two years. 19

20. Step 1: Significance test for π 2 - π 1 The parameter of interest is π 2 - π 1 Assumptions: –the sample is drawn from a random sample of some sort, –the parameter of interest is a variable with an interval scale, –the sample size is large enough that the sampling distribution of Pi hat2 – Pi hat1 is approximately normal. –The two samples are drawn independently 20

21. Step 2: Significance test for π 2 - π 1 The null hypothesis will be that there is no difference between the population proportions. This means that any difference we observe is due to random chance. H o: π 2 - π 1 = 0 (State an alpha here if you want to.) 21

22. Step 3: Significance test for π 2 - π 1 The test statistic has a standard form: z = (estimate of parameter – H o value of parameter) standard error of parameter Where pi hat is the overall weighted average –This means we are assuming equal variance in the two populations. –Q: why do we use an assumption of equal variance to estimate the standard error for the t-test? 22

23. Step 4: Significance test for π 2 - π 1 P-value of calculated z: Table A, or Stata: display 2 * (1 – normal(z) ), or Stata: testi (no data, just parameters) Stata: ttest (if data file in memory) 23

24. Step 5: Significance test for π 2 - π 1 Conclusion: Compare the p-value from step 4 to the alpha level in step 1. If p < α, reject H 0 If p ≥ α, do not reject H 0 State a conclusion about the statistical significance of the test. Briefly discuss the substantive importance of your findings. 24

25. Significance test for π 2 - π 1 : Example 1.Assumptions: random sample, interval-scale variable, sample size large enough that the sampling distribution of  2 -  1 is approximately normal, independent groups 2.Hypothesis: H o : π 2 - π 1 = 0 3.Test statistic: z = (122/345 – 268/1900) / SQRT[(390/2245)*(1 - 390/2245)*(1/345 + 1/1900)] = 9.59 4.p-value: p<<.001 5.conclusion: a.reject H 0 : attitudes were clearly different in 1994 than in 1982. b.furthermore, the observed difference of.21 is a substantively important change in attitudes. 25

26. Comparisons of two independent population proportions: Confidence Interval confidence interval: Notice that there is no overall weighted average Pi hat, as there is in a significance test for proportions. –Instead, we estimate two separate variances from the separate proportions. –Why? 26

27. STATA: Significance test for π 2 - π 1 : immediate command. prtesti 345.3536 1900.1411 STATA needs the following information: –sample size for group 1 (n = 345) –proportion for group 1 (p = 122/345) –sample size for group 2 (n = 1900) –proportion for group 2 (p = 268/1900) 27

28. STATA: Significance test for π 2 - π 1 : immediate command. prtesti 345.3536 1900.1411 Two-sample test of proportion x: Number of obs = 345 y: Number of obs = 1900 ------------------------------------------------------------------------------ Variable | Mean Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- x |.3536.0257393.3031518.4040482 y |.1411.0079865.1254467.1567533 -------------+---------------------------------------------------------------- diff |.2125.0269499.1596791.2653209 | under Ho:.0221741 9.58 0.000 ------------------------------------------------------------------------------ Ho: proportion(x) - proportion(y) = diff = 0 Ha: diff 0 z = 9.583 z = 9.583 z = 9.583 P |z| = 0.0000 P > z = 0.0000 Note the use of one standard error (unequal variance) for the confidence interval, and another (equal variance) for the significance test. 28

29. STATA command for a data set (#1). prtest nonstandard if (RACECEN1==1 | RACECEN1==2), by(RACECEN1) Two-sample test of proportion 1: Number of obs = 1389 2: Number of obs = 260 ------------------------------------------------------------------------------ Variable | Mean Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- 1 |.2800576.0120482.2564436.3036716 2 |.3538462.0296544.2957247.4119676 -------------+---------------------------------------------------------------- diff | -.0737886.0320084 -.1365239 -.0110532 | under Ho:.0307147 -2.40 0.016 ------------------------------------------------------------------------------ diff = prop(1) - prop(2) z = -2.4024 Ho: diff = 0 Ha: diff 0 Pr(Z z) = 0.9919 29

30. STATA command for a data set (#1). gen byte wrkslf0=wrkslf-1 (152 missing values generated). prtest wrkslf0 if wrkstat==1, by(sex) Two-sample test of proportion male: Number of obs = 874 female: Number of obs = 743 ------------------------------------------------------------------------------ Variable | Mean Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- male |.8272311.0127876.8021678.8522944 female |.9044415.0107853.8833027.9255802 -------------+---------------------------------------------------------------- diff | -.0772103.0167286 -.1099978 -.0444229 | under Ho:.0171735 -4.50 0.000 ------------------------------------------------------------------------------ diff = prop(male) - prop(female) z = -4.4959 Ho: diff = 0 Ha: diff 0 Pr(Z z) = 1.0000 30