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)
• Power1-beta

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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.
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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

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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

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Introduction to Biostatistics (PUBHLTH 540)
Hypothesis Testing-General Idea
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Introduction to Biostatistics (PUBHLTH 540)
Hypothesis Testing-General Idea
z8.08
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Introduction to Biostatistics (PUBHLTH 540)
Hypothesis Testing-General Idea
z8.3
z8.32
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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

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Introduction to Biostatistics (PUBHLTH 540)
Hypothesis Testing-General Idea
Null Hypothesis
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Introduction to Biostatistics (PUBHLTH 540)
Hypothesis Testing-General Idea
AlternativeHypothesis
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Introduction to Biostatistics (PUBHLTH 540)
Hypothesis Testing-General Idea
Critical region
2-sided test
Unusual
Unusual
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Introduction to Biostatistics (PUBHLTH 540)
Hypothesis Testing-General Idea
Critical region
1-sided test
Unusual
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Introduction to Biostatistics (PUBHLTH 540)
Hypothesis Testing-General Idea
2-sided test
Unusual
Unusual
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Introduction to Biostatistics (PUBHLTH 540)
Hypothesis Testing-General Idea
1-sided test
Unusual
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Introduction to Biostatistics (PUBHLTH 540)
Power of a Test-General Idea
1-sided test
Unusual
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Introduction to Biostatistics (PUBHLTH 540)
Power of a Test-General Idea
1-sided test
Unusual
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Introduction to Biostatistics (PUBHLTH 540)
Power of a Test-General Idea
1-sided test
1-sided test
Unusual
Unusual
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Introduction to Biostatistics (PUBHLTH 540)
Power of a Test-General Idea
1-sided test
1-sided test
Unusual
Unusual
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Introduction to Biostatistics (PUBHLTH 540)
Power of a Test-General Idea
1-sided test
1-sided test
1-sided test
Unusual
Unusual
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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.

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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.

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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.

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Introduction to Biostatistics (PUBHLTH 540)
Power- example

• Make a sketch.