Stat 529 (Winter 2011) Non-pooled t procedures (The Welch test) Reading: Section 4.3.2 The sampling distribution of Y 1 Y 2. An approximate sampling distribution for the t-ratio. The Sri Lankan analysis. Pooled or non-pooled? Caution: comparing population means when σ 1 σ 2. Performing a good analysis. Preliminary tests for normality and equal variances The modeling game plan. 1
The non-pooled t procedures The non-pooled t procedures or Welch test are available for performing statistical inference on µ 1 µ 2 when σ 1 σ 2. We assume 1. Y 11,..., Y 1n1 form a random sample from some population. Y 21,..., Y 2n2 form a random sample from some other population. 2. The two samples are independent of one another. 3. The population distributions are normal with unknown means µ 1 and µ 2, and with unknown standard deviations σ 1 and σ 2. As for the pooled setting, Y 1 Y 2 is an estimate of µ 1 µ 2. 2
The sampling distribution of Y 1 Y 2 From earlier we know that Y 1 Y 2 has a normal distribution with σ1 2 mean µ 1 µ 2 and S.D. + σ2 2. n 1 n 2 When σ 2 1 σ 2 2, the t-ratio, t = Y 1 Y 2 (µ 1 µ 2 ) s 2 1 + s2 2 n 1 n 2 does not exactly have a t distribution. 3
An approximate sampling distribution for Y 1 Y 2 However, the t-ratio does does have an approximate t distribution with df = ( ) s 2 2 1 + s2 2 n 1 n 2 (s 2 1/n 1 ) 2 (n 1 1) + (s2 2/n 2 ) 2 (n 2 1) degrees of freedom (this is called the Satterthwaite approximation). We use this as the basis of tests and confidence intervals for statistical inference on µ 1 µ 2. See the two sample non-pooled t/welch procedures handout. 4
The Sri Lankan analysis For the log 10 data, let us test H 0 : µ 1 = µ 2 versus H a : µ 1 µ 2, where µ 1 is the population mean log 10 zinc content found in rural Sri Lankan hair, and µ 2 is the population mean log 10 zinc content found in urban Sri Lankan hair. Here is part of the summaries of the log 10 data again: Descriptive Statistics: log10(zinc content) Population N N* Mean SE Mean StDev rural 15 0 2.7447 0.0888 0.3438 urban 11 0 2.966 0.102 0.339 5
The Sri Lankan analysis, continued 6
The Sri Lankan analysis, continued 7
Pooled or non-pooled? We use Stat Basic Statistics 2-Sample t in MINITAB to compare pooled and non-pooled tests. When we do not select Assume equal variance: Two-Sample T-Test and CI: log10(zinc content), Population Difference = mu (rural) - mu (urban) Estimate for difference: -0.221543 95% CI for difference: (-0.503048, 0.059962) T-Test of difference = 0 (vs not =): T-Value = -1.64 P-Value = 0.117 DF = 21 When we select Assume equal variance: Two-Sample T-Test and CI: log10(zinc content), Population Difference = mu (rural) - mu (urban) Estimate for difference: -0.221543 95% CI for difference: (-0.501565, 0.058479) T-Test of difference = 0 (vs not =): T-Value = -1.63 P-Value = 0.116 DF = 24 Both use Pooled StDev = 0.3418 8
Pooled or non-pooled, continued Advantages of pooled: Pooled test is slightly more powerful than non-pooled when σ 1 = σ 2 (especially when n 1 is much smaller than n 2, or vice versa). Model for pooled procedures is commonly used in other statistical procedures such as analysis of variance (ANOVA) or regression. Advantages of non-pooled: Robust to departures from the σ 1 = σ 2 assumption. A recommendation: 9
Caution: comparing population means when σ 1 σ 2 If two distributions have the same shape and spread then a difference of means is an adequate summary of the difference in the distributions. When the spread is different µ 1 µ 2 may be a bad summary of the difference in the distributions. 10
Performing a good analysis Bring all your subject matter knowledge to the table when you perform an analysis. Should σ 1 = σ 2? Should you expect populations to be normal? Or, are they normal after some transformation? Bring statistical knowledge to the table too! What weaknesses does the statistical procedure you are using have? Is it robust to the assumptions? Is it resistant? Does your experiment/study suggest weaknesses that may invalidate the analysis/model? 11
Preliminary tests for normality and equal variances A number of the t-procedures assume that the data are drawn from a normal population. The pooled t-procedures assume equality of variances. There are a number of tests in the literature for testing normality and equality of variance (i.e., σ 1 = σ 2 ). For example in testing for equality (homogeneity) of variance: Bartlett s test relies heavily on the assumption of normality even with large samples, the S.E. used in the test is wrong if populations are not normal. Levene s test is more robust to departures from normality. Some advice: 12
The modeling game plan 1. Look at your data (via graphical/numerical summaries). 2. Transform as necessary (for analysis or interpretation). 3. Analyze with or without outliers. 4. Report conclusions. NOTE: Outliers are local features of the data. A transformation (a global feature) can remove the problem of outliers. Always do 2. before 3. 13