Week Stat651
5 Feb 1014, 2003
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This Week's Reading
Chapter 6 8.1, 8.2
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The essential issue is that you have two populations to compare You compare them on the basis of a single variable You can compare their means, or their variability.
Week 4 Review: Comparing Two Populations
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Week 4 Review: Paired Comparisons
We have shown how to test
: population mean difference = 0;
: population mean difference 0;
using tstatistics (confidence intervals and tests).
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Week 4 Review: Paired Comparisons
There are two alternative methods that are not so affected by outliers These are the Wilcoxon signed rank test and the sign test
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There are two populations
Week 4 Review: Comparing Two Populations
Take a sample from each population The sample sizes need not be the same Population 1: Population 2:
n1 n2
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Week 4 Review: Comparing Two Populations How do we compare the population 1 means and ???? 2
The usual way is to take their difference: = -
1 2
If the population means are equal, what is their difference?
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The CI can of course be used to test hypotheses This is the same as
Week 4 Review: Comparing Two Populations
H 0 : 1 = 2 vs H a : 1
2
0
H 0 : 1 - 2 =0 vs H a : 1 - 2
So we just need to check whether 0 is in the interval, just as we have done
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1. Comparing Two Population Means
The data:
The populations: The aim:
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1. Comparing Two Populations
The aim: CI for
1 - 2
The estimate of the difference in population means is the This is a random variable: it has sample to sample variability
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1. Comparing Two Populations
Difference of sample means
X1 - X 2
"Population" mean from repeated sampling is -
1 2
The s.d. for the population is, which is unknown
+ n1 n 2
2 1
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2 2
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1. Comparing Two Populations
If you can reasonably believe that the population sd's are nearly equal, it is customary to pick the equal variance assumption and estimate the common standard deviation by
(n1 - 1)s + (n 2 - 1)s sp = n1 + n 2 - 2
2 1
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2 2
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1. Comparing Two Populations
The standard error of is the X1 - X 2 value
sp
1 1 + n1 n 2
The number of degrees of freedom is
n1 + n 2 - 2
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1. Comparing Two Populations
1 - 2 A (1-) 100% CI for is
t /2 (n1 +n 2 -2)s p 1 1 + n1 n 2
X1 - X 2
Note how the sample sizes determine the CI length
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1. Comparing Two Populations
Generally, you should make your sample sizes nearly equal, or at least not wildly unequal. Consider a total sample size of 100
X1 - X 2
t /2 (n1 +n 2 -2)s p
1 1 + n1 n 2
1 1 = 1 if n1 = 1, n2 = 99 + n1 n2
= 0.20 if n1 = 50, n2 = 50 Thus, in the former case, your CI would be
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1. Comparing Two Populations
The CI can of course be used to test hypotheses This is the same as
H 0 : 1 = 2 vs H a : 1
2
So we just need to check whether 0 is in the interval, just as we have done
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1. Comparing Two Populations: The t test
H 0 : 1 - 2 =0 vs H a : 1 - 2
0
There is something called a ttest, which gives you the information as to whether 0 is in the CI. It does not tell you where the means lie however, so it is of limited use. Pvalues tell you the same thing.
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1. Comparing Two Populations: The t test The tstatistic is defined by
X1 - X 2 t= 1 1 sp + n1 n 2
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1. Comparing Two Populations: The t test
You reject equality of means if
|t| > t /2 (n1 +n 2 -2)
In this case, is p < or is p > ?
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The question is whether arsenic ingestion is related to squamous cell carcinoma
2. Arsenic and Squamous Cell Skin Cancer
We used the transformation X = log(0.005 + toe arsenic)
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2. Arsenic and Squamous Cell Skin Cancer
0. 00
15%
Healthy
Percent
10%
5%
0%
1. 00
15%
Percent
10%
Cancer
5%
0% -5.00
-3.00
-1.00
1.00
log(0.005 + Toe Arsenic Level)
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2. Arsenic and Squamous Cell Skin Cancer
2
1
1
0
3 5 7 12 13 14 18 20
-1
2 4 6 9 8 10 11 15 16 17 19 21
log(0 .005 + Toe Arsenic)
-2
-3
804 805 808
-4 -5
N=
806 807
284
524
Cancer
Healthy
Squamous Cell Carcinoma Status
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2. Arsenic and Squamous Cell Skin Cancer, Healthy Cases
Normal Q-Q Plot of log(0.005 + Toe Arsenic)
0
-1
-2
Expected Normal Value
-3
-4
-5 -5 -4 -3 -2 -1 0
Observed Value
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2. Arsenic and Squamous Cell Skin Cancer: Cancer Cases
Normal Q-Q Plot of log(0.005 + Toe Arsenic)
0
-1
-2
Expected Normal Value
-3
-4
-5 -5 -4 -3 -2 -1 0 1
Observed Value
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Healthy, s = 0.5838, IQR = 0.6913
2. Arsenic and Squamous Cell Skin Cancer
Squamous, s = 0.6629, IQR = 0.7582 These statistics and box plots indicate that the two populations do not have vastly different variabilities.
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Healthy: mean = 2.3057, n = 524, se = 0.02551 Squamous: mean = 2.3365, n = 284, se = 0.03934 Mean difference = 0.03077, se = 0.04515 95% CI= [0.0579, 0.1194] p = 0.496: what does this mean?
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2. Arsenic and Squamous Cell Skin Cancer
Because sample means and standard deviations are sensitive to outliers, so to are comparisons of populations based on them Rank tests form a robust alternative, that can be used to check the results of tstatistic inferences You are looking for major discrepancies, and then trying to explain them
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3.Robust Inference via Rank Tests 3.1 Motivation
Rank tests are very easy to compute, and SPSS provide them. The algorithm is to assign ranks to each observation in the pooled data set Then apply a ttest to these ranks Robust because ranks can never get wild
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3. Robust Inference via Rank Tests 3.2 Method
Typically called the Wilcoxon rank sum test
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Here is how data are ranked
3. Robust Inference via Rank Tests 3.3 Numerical Example
22 50 45 55 81 56
Data #1 3 7 #2 28 44
Ranks #1 1 2 3 6 10
#2 4 5 7 8 9 Now run a ttest
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The rank tests give the same answer no matter whether you take the raw data, their logarithms or their square roots.
3. Robust Inference via Rank Tests 3.4 Properties
If you have data (raw or transformed) that pass qq plots tests, then Wilcoxon and ttest should have much the same pvalues In this case, you can use the latter to get CI's
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In SPSS, you can get Wilcoxon rank sum tests as follows (SPSS calls them Mann Whitney U) "Analyze", "Nonparametric Tests", "2 independent samples"
3. Robust Inference via Rank Tests 3.5 SPSS
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3. Robust Inference via Rank Tests 3.6 Example
Illustrate with NHANES to look at p-values for saturated fat and its logarithm Saturated Fat pvalues: ttest = 0.057 , rank test = 0.014 Log(Saturated Fat): ttest = 0.012, rank test = 0.014 Note how the transform, which is more bell shaped, agrees more closely with the rank test!
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We now turn to making inferences when there are 3 or more This populations is classically called Analysis of Variance, or ANOVA It is somewhat more formula dense than what we have been used to. Tests for normality are also somewhat more complex
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4 How About More Than Two Populations?
4.1 ANOVA Introduction
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.
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4.1 ANOVA Introduction
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. In general, if there are t populations, there are t(t1)/2 twosample comparisons.
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4.1 ANOVA Introduction
The controversy revolves around the concept of Type I error Specifically, if we do six 95% confidence intervals, what is the probability that at least one of them does not include the true mean? It is higher than 5%!
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4.1 ANOVA Introduction
One school of thought is to stick to a few major hypotheses (24 say), do 95% CI on them, and label anything else as exploratory, with possibly inflated Type I errors Another school thinks that you should control the chance of making any errors at all to be 5% 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.
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4.1 ANOVA Introduction
One procedure that is often followed is to do a preliminary test to see whether there are any differences among the populations Then, once you conclude that some differences exist, you allow somewhat more informality in deciding where those differences manifest themselves The first step is the ANOVA Ftest
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4.2 ANOVA Notations
The Ftest is easy to compute, and provided in all statistical packages The populations are 1, 2, ..., t
n1 ,L, n t The population means are 1 ,L , t
The sample sizes are The hypothesis to test is
H 0 : 1 = 2 L = t
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4.2 ANOVA Notations
The data from population i are The sample mean from population i is The sample mean of all the data is
Yi1 , Yi2 ,L , Yin i
Y iC
YC
The total sample size is the total number of observations, called n
T
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4.3 ANOVA Table
The ANOVA Table (demo in SPSS in class of ACS data) "Analyze" "Compare Means" "1way ANOVA": I'll now show you what each item is
ANOVA
Baseline FFQ Sum of Squares 960.287 15275.639 16235.925 df 2 181 183 Mean Square 480.143 84.396 F 5.689 Sig. .004
Between Groups Within Groups Total
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4.3 ANOVA Table
The sample mean from population i is The sample mean of all the data is
Y iC
The idea of the Ftest is based on distances The distance of the data to the overall mean is 2
YC
TSS =
(Yij - Y C )
ij
TSS = Total Sum of Squares
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4.3 ANOVA Table
nT -1 This has degrees of freedom
So, the total sample size is the degrees of freedom for TSS + 1
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4.3 ANOVA Table
Next comes the "Between Groups" row
ANOVA Baseline FFQ Sum of Squares 960.287 15275.639 16235.925 df 2 181 183 Mean Square 480.143 84.396 F 5.689 Sig. .004
Between Groups Within Groups Total
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4.3 ANOVA Table
The sum of squares between groups is
n i (Y iC - Y )
i
2
It has t1 degrees of freedom, so the number of populations is the degrees of freedom between groups + 1.
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4.3 ANOVA Table
Next comes the "Within Groups" row
ANOVA Baseline FFQ Sum of Squares 960.287 15275.639 16235.925 df 2 181 183 Mean Square 480.143 84.396 F 5.689 Sig. .004
Between Groups Within Groups Total
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4.3 ANOVA Table
The distance of the observations to their sample means is
SSW =
ij
(Yij - Y iC)
2
This is the Sum of Squares Within
nT - t It has degrees of freedom
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4.3 ANOVA Table
Next comes the "Mean Squares" These are the different sums of squares divided by their degrees of freedom
ANOVA
Baseline FFQ Sum of Squares 960.287 15275.639 16235.925 df 2 181 183 Mean Square 480.143 84.396 F 5.689 Sig. .004
Between Groups Within Groups Total
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4.3 ANOVA Table
Next comes the Fstatistic It is the ratio of th...