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STAT 651 Lecture # 11 Copyright (c) Bani Mallick 1 Topics in Lecture #11 Comparing two population means Review to date Copyright (c) Bani Mallick 2 Book Sections Covered in Lecture #11 All previous chapters Copyright (c) Bani Mallick 3 Lecture 10 Review: Robust Inference via Rank Tests Because sample means and sd’s are sensitive to outliers, so too are comparisons of populations based on them Rank tests form a robust alternative, that can be used to check the results of t-statistic inferences You are looking for major discrepancies, and then trying to explain them Copyright (c) Bani Mallick 4 Lecture 10 Review: Robust Inference via Rank Tests Typically called the Wilcoxon rank sum test The algorithm is to assign ranks to each observation in the pooled data set Then apply a t-test to these ranks Robust because ranks can never get wild Copyright (c) Bani Mallick 5 Lecture 10 Review: Robust Inference via Rank Tests The rank tests give the same answer no matter whether you take the raw data, their logarithms or their square roots. If you have data (raw or transformed) that pass q-q plots tests, then Wilcoxon and t-test should have much the same p-values In this case, you can use the latter to get CI’s Copyright (c) Bani Mallick 6 Lecture 10 Review: Inference for Equality of Variances SPSS uses what is called Levene’s test From the SPSS Help file (slightly edited) Levene Test For each case, it computes the absolute difference between the value of that case and its cell mean and performs a t-test on those absolute differences. Copyright (c) Bani Mallick 7 Inference When the Variances Are Not Equal Generally, you should try to find a scale (log, square root) for which the variances are approximately equal. If you cannot, then a small change is needed in the confidence intervals and test statistics. Copyright (c) Bani Mallick 8 The Confidence Interval With Unequal Variances The CI for μ -μ is given by 1 2 2 1 2 2 s s X1 -X 2 ±t α/2 (n1 + n 2 -2) + n1 n 2 Copyright (c) Bani Mallick 9 The t-test Statistic With Unequal Variances The t-statistic is defined by t= X1 X 2 2 1 2 2 s s n1 n 2 Copyright (c) Bani Mallick 10 The Aortic Stenosis Data: A Review Let’s remember that the aortic stenosis data is comparing healthy with sick kids Aortic Valve Area (AVA) and Body Surface Area (BSA) are noninvasive measures The idea is to see if we can use these measures to understand whether a kid is stenotic, without having to do an invasive test Or at least to be able to narrow down those who need an invasive test Copyright (c) Bani Mallick 11 The Aortic Stenosis Data: A Review We will study the variable log(1 + AVA/BSA) I’ll try to review much of what we have done to this point by using these data There was a kids with an enormous AVA: I’ll delete this kid from all analyses First comes the relative frequency histogram Copyright (c) Bani Mallick 12 The Aortic Stenosis Data: A Review First comes the relative frequency histogram The height of each bar is the % of the sample which lies in the interval for that bar Copyright (c) Bani Mallick 13 The Aortic Stenosis Data: A Review: what % have X > 1? Copyright (c) Bani Mallick 14 The Aortic Stenosis Data: A Review Next comes the boxplot to Compare medians Compare variability Screen for massive outliers Copyright (c) Bani Mallick 15 The Aortic Stenosis Data: A Review: read off median, IQR, outliers 1.4 1.2 1.0 99 72 79 .8 .6 .4 .2 0.0 1 -.2 N= 70 55 Healthy Stenoti Health Status Copyright (c) Bani Mallick 16 The Aortic Stenosis Data: A Review Next comes the q-q plot to check for approximate bell-shape Why does it matter that the data are approximately bell-shaped? First: statistical inferences such as confidence intervals are most accurate and most powerful Second: Probability calculations are most accurate Copyright (c) Bani Mallick 17 The Aortic Stenosis Data: A Review For the healthy kids: what do you think? Normal Q-Q Plot of Log(1 + (AVA to BSA Ratio)) 1.4 1.2 Expected Normal Value 1.0 .8 .6 .4 .2 -.2 0.0 .2 .4 .6 .8 1.0 1.2 1.4 Observed Value Copyright (c) Bani Mallick 18 The Aortic Stenosis Data: A Review For the stenotic kids: what do you think? Normal Q-Q Plot of Log(1 + (AVA to BSA Ratio)) 1.0 .8 Expected Normal Value .6 .4 .2 0.0 0.0 .2 .4 .6 .8 1.0 Observed Value Copyright (c) Bani Mallick 19 The Aortic Stenosis Data: A Review Next we do some simple summary statistics Healthy: n = 70, mean = 0.84, median = 0.83, sd = 0.22, iqr = 0.31, se (of mean) = 0.026 Stenotic: n = 55, mean = 0.47, median = 0.46, sd = 0.17, iqr = 0.22, se (of mean) = 0.023 What does the IQR mean? Copyright (c) Bani Mallick 20 The Aortic Stenosis Data: A Review Healthy: n = 70, mean = 0.84, median = 0.83, sd = 0.22, iqr = 0.31, se (of mean) = 0.026 Degrees of freedom = 70 – 1 = 69 95% CI for population mean (from SPSS) = 0.79 to 0.89 Interpret this! Copyright (c) Bani Mallick 21 The Aortic Stenosis Data: A Review Healthy: n = 70, mean = 0.84, median = 0.83, sd = 0.22, iqr = 0.31, se (of mean) = 0.026 Estimate the probability that a healthy child has a log(1+ASA/BSA) less that 0.46 Pr(X < 0.46): z-score = (0.46 – 0.84)/0.22 = -1.73 Table 1: Pr(X < 0.46) ~ 0.0418 Interpret this! Copyright (c) Bani Mallick 22 The Aortic Stenosis Data: A Review Note: for computing confidence intervals for the population mean I use the standard error In computing probabilities about the population I use the standard deviation Which one is affected by the sample size? Copyright (c) Bani Mallick 23 The Aortic Stenosis Data: A Review Next I will compare the population means We have concluded that the data are fairly (although not exactly) normally distributed We have also concluded that the variabilities are not too awfully different, although the stenotic kids appear to be less variable. So, we will try out the t-test type inference Copyright (c) Bani Mallick 24 The Aortic Stenosis Data: A Review SPSS computes a confidence interval for the difference in the population means H0 :μ1 -μ 2 =0 vs Ha :μ1 -μ 2 0 We conclude the population means are different if the confidence interval does not include zero Copyright (c) Bani Mallick 25 The Aortic Stenosis Data: A Review SPSS Group Statistics Log(1 + (AVA to BSA Ratio)) Health Status Healthy Stenotic N 70 55 Mean .8424 .4659 Std. Error Std. Deviation Mean .2199 2.628E-02 .1734 2.338E-02 Copyright (c) Bani Mallick 26 The Aortic Stenosis Data: A Review SPSS Independent Samples Test Levene's Test for Equality of Variances F Log(1 + (AVA to BSA Ratio)) Equal variances ass umed Equal variances not as sumed 3.346 Sig. .070 t-tes t for Equality of Means t df Sig. (2-tailed) Mean Difference Std. Error Difference 95% Confidence Interval of the Difference Lower Upper 10.407 123 .000 .3765 3.618E-02 .3049 .4482 10.705 122.997 .000 .3765 3.518E-02 .3069 .4462 Copyright (c) Bani Mallick 27 The Aortic Stenosis Data: A Review The 95% confidence interval is from 0.30 to 0.45 Interpret this! Copyright (c) Bani Mallick 28 The Aortic Stenosis Data: A Review The p-value is 0.000. Interpret this! What would have happened if I had done a 99% CI? Copyright (c) Bani Mallick 29 The Aortic Stenosis Data: A Review The p-value is 0.000. Interpret this! What would have happened if I had done a 99% CI? It would not have covered zero, and I would have rejected the null hypothesis that the two population means are the same. Copyright (c) Bani Mallick 30 The Aortic Stenosis Data: A Review Outliers can really mess up an analysis We use the Wilcoxon rank sum test for this There were no massive outliers, the data were nearly normal, so we expect to get just about the same p-value It is 0.000, just as for the CI analysis Test Statisticsa Mann-Whitney U Wilcoxon W Z Asymp. Sig. (2-tailed) Log(1 + (AVA to BSA Ratio)) 305.500 1845.500 -8.055 .000 a. Grouping Variable: Health Status Copyright (c) Bani Mallick 31 The Aortic Stenosis Data: A Review Next we will use Levene’s test to compare variability SPSS has its version of Levene’s test, which is a t-test on the absolute differences with the sample mean: the p-value for this is 0.07 Weak evidence of differences in variability Copyright (c) Bani Mallick 32 ANalysis Of VAriance We now turn to making inferences when there are 3 or more populations This is classically called ANOVA It is somewhat more formula dense than what we have been used to. Tests for normality are also somewhat more complex Copyright (c) Bani Mallick 33 ANOVA The Analysis of Variance is often known as ANOVA We are going to consider its simplest form, namely comparing 3 or more populations. If there are two populations, we have covered this in the first part of the course Confidence intervals for the difference in two population means Wilcoxon rank sum test. Copyright (c) Bani Mallick 34 ANOVA Suppose we form three populations on the basis of body mass index (BMI): BMI < 22, 22 <= BMI < 28, BMI > 28 This forms 3 populations We want to know whether the three populations have the same mean caloric intake, or if their food composition differs. Copyright (c) Bani Mallick 35 ANOVA Concho water snake data for males have four separate subpopulations, depending on the year class. Are there differences in snout-to-vent lengths among these populations? Copyright (c) Bani Mallick 36 ANOVA As in all analyses, we will combine graphical analyses, summary statistics and formal hypothesis tests and confidence intervals to for a picture of what the data are telling us Copyright (c) Bani Mallick 37 ANOVA There is considerable controversy as to how to compare 3 or more populations. One possibility is to compare them 2 at a time In the body mass example, there are 3 such comparisons Low BMI versus Middle BMI Low BMI versis High BMI Middle BMI versus High BMI Copyright (c) Bani Mallick 38 ANOVA In the Concho water snake example there are 6 such comparisons. 1 year olds versus 2 year olds 1 year olds versus 3 year olds 1 year olds versus 4 year olds 2 year olds versus 3 year olds 2 year olds versus 4 year olds 3 year olds versus 4 year olds Copyright (c) Bani Mallick 39 ANOVA In general, if there are t populations, there are t(t-1)/2 two-sample comparisons. Special Note: t is a symbol used in the book to denote the number of populations Copyright (c) Bani Mallick 40 ANOVA In general, if there are t populations, there are t(t-1)/2 two-sample comparisons. The controversy revolves around the concept of Type I error Specifically, if we do 6 different 95% confidence intervals, what is the probability that one or more of them do not include the true mean? Certainly, it is higher than 5%! Copyright (c) Bani Mallick 41 ANOVA If you do lots of 95% confidence intervals, you’d expect by chance that about 5% will be wrong Thus, if you do 20 confidence intervals, you expect 1 = 20 x 5% will not include the true population parameter This is a fact of life Copyright (c) Bani Mallick 42 ANOVA If you do lots of 95% confidence intervals, you’d expect by chance that about will be wrong One school of thought is to stick to a few major hypotheses (2-4 say), do 95% CI on them, and label anything else as exploratory, with possibly inflated Type I errors I am of this school Copyright (c) Bani Mallick 43 ANOVA If you do lots of 95% confidence intervals, you’d expect by chance that about 5% will be wrong Another school thinks that you should control the chance of making any errors at all to be 5% I am NOT of this school. Why not: am I just crazy? Copyright (c) Bani Mallick 44 ANOVA Another school thinks that you should control the chance of making any errors at all to be 5% I am NOT of this school. I worry about Type II error (power). To be 95% confident that every single one of my conclusions is correct, I will not have much power to detect meaningful changes or differences. Copyright (c) Bani Mallick 45