introduction to biostatistics (pubhlth 540) hypothesis testing
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Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing. General Idea How unusual is the result? Test statistics Type I error (alpha level) p-value Type II error (beta level) Power=1-beta. Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea. - PowerPoint PPT PresentationTRANSCRIPT
1
Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing
• General Idea
• How unusual is the result?
• Test statistics
• Type I error (alpha level)
• p-value
• Type II error (beta level) – Power=1-beta
2
Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea
• Total cholesterol (mg/dl) is measured on a simple random sample of 32 women over the age of 60. Is there evidence that mean cholesterol is different in women of this age group as compared with women under age 50?
• Estimate of TC: (see ejs09b540p36.sas)
32n 244Y 34S • Plot histogram of SRS of n=32 from women <50. (see
ejs09b540p37.sas)Figure 1. Histogram of sample means of n=32 for tc from Population of N=93
Source: ejs09b540p37.sas 10/27/2009 by ejs
170 173 176 179 182 185 188 191 194 197 200 203 206 209 212 215 218 221 224 227 230
0
5
10
Pe
rce
nt
mn_samp
244
Result is very unusual relative to what we’d expect from sampling.
Conclude the mean is differnet.
3
Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea
32n244Y 34S
Figure 1. Histogram of sample means of n=32 for tc from Population of N=93
Source: ejs09b540p37.sas 10/27/2009 by ejs
170 173 176 179 182 185 188 191 194 197 200 203 206 209 212 215 218 221 224 227 230
0
5
10
Pe
rce
nt
mn_samp
244
• Histogram of distribution of sample means/standardized value-– need to know mean and variance of TC for women < 50.
– use Z if variance is known, t if variance is estimated
0cal
y
yz
0cal
y
yt
s
• for women < 50:
4
Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea
32n244Y 34S
244
• Histogram of distribution of sample means
• or … histogram of standardized values of the difference of the sample mean from the mean TC for women < 50– need to know mean and variance of TC for women < 50.
– use Z if variance is known, t if variance is estimated
0cal
y
yz
0cal
y
yt
s
• for women < 50: 0 194
35
244 194
35 / 328.08
calz
244 194
34 / 328.32
calt
5
Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea
Figure 1. Histogram of sample means of n=32 for tc from Population of N=93
Source: ejs09b540p37.sas 10/27/2009 by ejs
170 173 176 179 182 185 188 191 194 197 200 203 206 209 212 215 218 221 224 227 230 233 236 239 242 245
0
5
10
Pe
rce
nt
mn_samp 244
6
Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea
Figure 2. Histogram of sample z-scores of n=32 for tc from Population of N=93
Source: ejs09b540p37.sas 10/28/2009 by ejs
-4.0 -3.4 -2.8 -2.2 -1.6 -1.0 -0.4 0.2 0.8 1.4 2.0 2.6 3.2 3.8 4.4 5.0 5.6 6.2 6.8 7.4 8.0 8.6
0
5
10 P
erc
en
t
z
z=8.08
7
Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea
z=8.3
Figure 3. Histogram of sample t-scores of n=32 for tc from Population of N=93
Source: ejs09b540p37.sas 10/28/2009 by ejs
-4.0 -3.4 -2.8 -2.2 -1.6 -1.0 -0.4 0.2 0.8 1.4 2.0 2.6 3.2 3.8 4.4 5.0 5.6 6.2 6.8 7.4 8.0 8.6
0
5
10
Pe
rce
nt
t
z=8.32
8
Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea
• Is the result ‘unusual’?– Decide a level of ‘unusualness’– usually set at values so that 5% of time, sample mean
would be further away
– If in either direction, then 2.5% on either side, and test is called ‘2-sided’
– If unusual is important only in one direction (drug lowers cholesterol), then put all 5% on one side
– Null hypothesis is ‘usual’ or commonly accepted position.
– Alternative hypothesis is what you want to ‘prove’
0.05level
0.05level
9
Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea
Figure 1. Histogram of sample means of n=32 for tc from Population of N=93
Source: ejs09b540p37.sas 10/27/2009 by ejs
170 173 176 179 182 185 188 191 194 197 200 203 206 209 212 215 218 221 224 227 230 233 236 239 242 245
0
5
10
Per
cent
mn_samp
0
Null Hypothesis:
0 0:H
10
Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea
Figure 1. Histogram of sample means of n=32 for tc from Population of N=93
Source: ejs09b540p37.sas 10/27/2009 by ejs
170 173 176 179 182 185 188 191 194 197 200 203 206 209 212 215 218 221 224 227 230 233 236 239 242 245
0
5
10
Per
cent
mn_samp
0
AlternativeHypothesis
0 0:H
1 0:H
11
Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea
Figure 1. Histogram of sample means of n=32 for tc from Population of N=93
Source: ejs09b540p37.sas 10/27/2009 by ejs
170 173 176 179 182 185 188 191 194 197 200 203 206 209 212 215 218 221 224 227 230 233 236 239 242 245
0
5
10
Per
cent
mn_samp
0
2-sided test
0 0:H
1 0:H
UnusualUnusual
12
Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea
Figure 1. Histogram of sample means of n=32 for tc from Population of N=93
Source: ejs09b540p37.sas 10/27/2009 by ejs
170 173 176 179 182 185 188 191 194 197 200 203 206 209 212 215 218 221 224 227 230 233 236 239 242 245
0
5
10
Per
cent
mn_samp
0
1-sided test
0 0:H
1 0:H
Unusual
13
Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea
Figure 1. Histogram of sample means of n=32 for tc from Population of N=93
Source: ejs09b540p37.sas 10/27/2009 by ejs
170 173 176 179 182 185 188 191 194 197 200 203 206 209 212 215 218 221 224 227 230 233 236 239 242 245
0
5
10
Per
cent
mn_samp
0
2-sided test
0 0:H
1 0:H
UnusualUnusual
y
2 ( )p value P Y y
14
Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing-General Idea
Figure 1. Histogram of sample means of n=32 for tc from Population of N=93
Source: ejs09b540p37.sas 10/27/2009 by ejs
170 173 176 179 182 185 188 191 194 197 200 203 206 209 212 215 218 221 224 227 230 233 236 239 242 245
0
5
10
Per
cent
mn_samp
0
1-sided test
0 0:H
1 0:H
Unusual
y
( )p value P Y y
15
Introduction to Biostatistics (PUBHLTH 540) Power of a Test-General Idea
Figure 1. One Sided test for a particular alternative and Histogram of sample means assuming null hypothesis is true based on of n=32 for tc from Population of N=93
Source: ejs09b540p37.sas 10/27/2009 by ejs
170 173 176 179 182 185 188 191 194 197 200 203 206 209 212 215 218 221 224 227 230 233 236 239 242 245
0
5
10
Per
cent
mn_samp
0
1-sided test
0 0:H
1 : 227H
Unusual
y
( )p value P Y y
16
Introduction to Biostatistics (PUBHLTH 540) Power of a Test-General Idea
Figure 1. One Sided test for a particular alternative and Histogram of sample means assuming null hypothesis is true based on of n=32 for tc from Population of N=93
Source: ejs09b540p37.sas 10/27/2009 by ejs
170 173 176 179 182 185 188 191 194 197 200 203 206 209 212 215 218 221 224 227 230 233 236 239 242 245
0
5
10
Per
cent
mn_samp
0
1-sided test
0 0:H
1 : 227H
Unusual
y
( )p value P Y y
17
Introduction to Biostatistics (PUBHLTH 540) Power of a Test-General Idea
0
1-sided test
0 0:H
1 : 227H
Unusual
y
( )p value P Y y
Figure 1. Histogram of sample means of n=32 for tca from Population of N=93
Source: ejs09b540p37a.sas 11/16/2010 by ejs
170 173 176 179 182 185 188 191 194 197 200 203 206 209 212 215 218 221 224 227 230 233 236 239 242 245
0
5
10
Pe
rce
nt
mn_samp
Unusual
0
1( 206 | 227)
0.999
Power P Y
1 227
1-sided test
0 0:H
1 : 227H
18
Introduction to Biostatistics (PUBHLTH 540) Power of a Test-General Idea
0
1-sided test
0 0:H
1 : 227H
Unusual
y
( )p value P Y y
Figure 1. Histogram of sample means of n=32 for tca from Population of N=93
Source: ejs09b540p37a.sas 11/16/2010 by ejs
170 173 176 179 182 185 188 191 194 197 200 203 206 209 212 215 218 221 224 227 230 233 236 239 242 245
0
5
10
Pe
rce
nt
mn_samp
Unusual
0
1( 206 | 227)
0.999
Power P Y
1 227
1-sided test
0 0:H
1 : 227H
19
Introduction to Biostatistics (PUBHLTH 540) Power of a Test-General Idea
0
1-sided test
0 0:H
1 : 227H
Unusual
y
( )p value P Y y
Figure 1. Histogram of sample means of n=32 for tca from Population of N=93
Source: ejs09b540p37a.sas 11/16/2010 by ejs
170 173 176 179 182 185 188 191 194 197 200 203 206 209 212 215 218 221 224 227 230 233 236 239 242 245
0
5
10
Pe
rce
nt
mn_samp
Unusual
0
1( 206 | 227)
0.999
Power P Y
1-sided test
0 0:H
1 : 227H
Figure 1. Histogram of sample means of n=32 for tca from Population of N=93
Source: ejs09b540p37a.sas 11/16/2010 by ejs
170 173 176 179 182 185 188 191 194 197 200 203 206 209 212 215 218 221 224 227 230 233 236 239 242 245
0
5
10
Pe
rce
nt
mn_samp
0
1( 206 | 215)
0.962
Power
P Y
1 215
1-sided test
0 0:H
1 : 215H
1 0.962
0.034
Power
20
Introduction to Biostatistics (PUBHLTH 540) Power- example
• We know TC for males is normally distributed with mean 220, and variance 1524. Let us test the null hypothesis that the mean is 220, versus an alternative hypothesis that the mean is 205 based on a one sided test with n=25. What is the power of the test?
• Figure out the rejection region under the null hypothesis in terms of the distribution of sample means.
• Make a sketch.• Use the z-applet with an assumption that the
alternative hypothesis is true to figure the power.
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Introduction to Biostatistics (PUBHLTH 540) Power- example
• Figure out the rejection region under the null hypothesis in terms of the distribution of sample means.
00
1.645 0.05
1.645 1.645
39220 1.645
5
207
P Z
YP P Y
n n
P Y
P Y
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Introduction to Biostatistics (PUBHLTH 540) Power- example
• Figure out the rejection region under the null hypothesis in terms of the distribution of sample means.
23
Introduction to Biostatistics (PUBHLTH 540) Power- example
• Make a sketch. 0.6012Power