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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

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

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

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

Introduction to Biostatistics (PUBHLTH 540)

Hypothesis Testing-General Idea

244

Introduction to Biostatistics (PUBHLTH 540)

Hypothesis Testing-General Idea

z8.08

Introduction to Biostatistics (PUBHLTH 540)

Hypothesis Testing-General Idea

z8.3

z8.32

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

Introduction to Biostatistics (PUBHLTH 540)

Hypothesis Testing-General Idea

Null Hypothesis

Introduction to Biostatistics (PUBHLTH 540)

Hypothesis Testing-General Idea

AlternativeHypothesis

Introduction to Biostatistics (PUBHLTH 540)

Hypothesis Testing-General Idea

Critical region

2-sided test

Unusual

Unusual

Introduction to Biostatistics (PUBHLTH 540)

Hypothesis Testing-General Idea

Critical region

1-sided test

Unusual

Introduction to Biostatistics (PUBHLTH 540)

Hypothesis Testing-General Idea

2-sided test

Unusual

Unusual

Introduction to Biostatistics (PUBHLTH 540)

Hypothesis Testing-General Idea

1-sided test

Unusual

Introduction to Biostatistics (PUBHLTH 540)

Power of a Test-General Idea

1-sided test

Unusual

Introduction to Biostatistics (PUBHLTH 540)

Power of a Test-General Idea

1-sided test

Unusual

Introduction to Biostatistics (PUBHLTH 540)

Power of a Test-General Idea

1-sided test

1-sided test

Unusual

Unusual

Introduction to Biostatistics (PUBHLTH 540)

Power of a Test-General Idea

1-sided test

1-sided test

Unusual

Unusual

Introduction to Biostatistics (PUBHLTH 540)

Power of a Test-General Idea

1-sided test

1-sided test

1-sided test

Unusual

Unusual

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.

Introduction to Biostatistics (PUBHLTH 540)

Power- example

- Figure out the rejection region under the null

hypothesis in terms of the distribution of sample

means.

Introduction to Biostatistics (PUBHLTH 540)

Power- example

- Figure out the rejection region under the null

hypothesis in terms of the distribution of sample

means.

Introduction to Biostatistics (PUBHLTH 540)

Power- example

- Make a sketch.