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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) – PowerPoint PPT presentation

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Title: Introduction to Biostatistics (PUBHLTH 540) Hypothesis Testing


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)
  • Power1-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)
  • Plot histogram of SRS of n32 from women lt50.
    (see ejs09b540p37.sas)

Result is very unusual relative to what wed
expect from sampling. Conclude the mean is
differnet.
244
3
Introduction to Biostatistics (PUBHLTH 540)
Hypothesis Testing-General Idea
  • Histogram of distribution of sample
    means/standardized value-
  • need to know mean and variance of TC for women lt
    50.
  • use Z if variance is known, t if variance is
    estimated
  • for women lt 50

244
4
Introduction to Biostatistics (PUBHLTH 540)
Hypothesis Testing-General Idea
  • Plot Histogram of distribution of sample means
    under Null H
  • or histogram of standardized values of the
    difference of the sample mean from the mean TC
    for women lt 50
  • need to know mean and variance of TC for women lt
    50.
  • for women lt 50
  • use Z if variance is known, t if variance is
    estimated

244
5
Introduction to Biostatistics (PUBHLTH 540)
Hypothesis Testing-General Idea
244
6
Introduction to Biostatistics (PUBHLTH 540)
Hypothesis Testing-General Idea
z8.08
7
Introduction to Biostatistics (PUBHLTH 540)
Hypothesis Testing-General Idea
z8.3
z8.32
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 (also called TYPE 1
    Error)
  • If in either direction, then 2.5 on either side,
    and test is called 2-sided
  • Called 2-sided test
  • If unusual is important only in one direction
    (drug lowers cholesterol), then put all 5 on one
    side
  • Called 1-sided test
  • Null hypothesis is usual or commonly accepted
    position.
  • Alternative hypothesis is what you want to prove

9
Introduction to Biostatistics (PUBHLTH 540)
Hypothesis Testing-General Idea
Null Hypothesis
10
Introduction to Biostatistics (PUBHLTH 540)
Hypothesis Testing-General Idea
AlternativeHypothesis
11
Introduction to Biostatistics (PUBHLTH 540)
Hypothesis Testing-General Idea
Critical region
2-sided test
Unusual
Unusual
12
Introduction to Biostatistics (PUBHLTH 540)
Hypothesis Testing-General Idea
Critical region
1-sided test
Unusual
13
Introduction to Biostatistics (PUBHLTH 540)
Hypothesis Testing-General Idea
2-sided test
Unusual
Unusual
14
Introduction to Biostatistics (PUBHLTH 540)
Hypothesis Testing-General Idea
1-sided test
Unusual
15
Introduction to Biostatistics (PUBHLTH 540)
Power of a Test-General Idea
1-sided test
Unusual
16
Introduction to Biostatistics (PUBHLTH 540)
Power of a Test-General Idea
1-sided test
Unusual
17
Introduction to Biostatistics (PUBHLTH 540)
Power of a Test-General Idea
1-sided test
1-sided test
Unusual
Unusual
18
Introduction to Biostatistics (PUBHLTH 540)
Power of a Test-General Idea
1-sided test
1-sided test
Unusual
Unusual
19
Introduction to Biostatistics (PUBHLTH 540)
Power of a Test-General Idea
1-sided test
1-sided test
1-sided test
Unusual
Unusual
20
Introduction to Biostatistics (PUBHLTH 540)
Power- example
  • Assume in one population, we know TC for males is
    normally distributed with mean 220, and variance
    1524. Our interest is in mean TC for men in a
    different population. We would like to know
    whether TC is less in the other population (vs a
    null hypothesis that it is equal to or greater
    than 220). Consider a one sided test of the null
    hypothesis. Suppose we select a sample of n25
    subjects from the new population. 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 n25. 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 indicating the critical region (on
    the scale of TC).
  • Use the z-applet with an assumption that the
    alternative hypothesis is true to figure the
    power.

21
Introduction to Biostatistics (PUBHLTH 540)
Power- example
  • Figure out the rejection region under the null
    hypothesis in terms of the distribution of sample
    means.

22
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.
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