Copyright (c) Bani Mallick1 STAT 651 Lecture 7. Copyright (c) Bani Mallick2 Topics in Lecture #7...

Copyright (c) Bani Mallick 1

STAT 651

Lecture 7


Topics in Lecture #7 Sample size for fixed power

Never, ever, accept a null hypothesis

Paired comparisons in SPSS

Student’s t-distributions

Confidence intervals when is unknown

SPSS output on confidence intervals, without formulae


Book Sections Covered in Lecture #7

Chapter 5.5 (sample size)

Chapter 6.4 (paired data)

Chapter 5.7 (t-distribution)

My own screed (never, ever, accept a null hypothesis)


Lecture 6 Review: Hypothesis Testing

Suppose you want to know whether the population mean change in reported caloric intake equals zero

We have already done this!!!!!

Confidence intervals tell you where the population mean is, with specified probability

If zero is not in the confidence interval, then you can reject the hypothesis


Lecture 6 Review: Type I Error (False Reject)

A Type I error occurs when you say that the null hypothesis is false when in fact it is true

You can never know for certain whether or not you have made such an error

You can only control the probability that you make such an error

t is convention to make the probability of a Type I error 5%, although 1% and 10% are also used


Lecture 6 Review: Type I Error Rates

Choose a confidence level, call it 1 -

The Type I error rate is confidence interval: = 10%

confidence interval: = 5%

confidence interval: = 1%


Lecture 6 Review: Type II: The Other Kind of Error

The other type of error occurs when you do NOT reject even though it is false

This often occurs because you study sample size is too small to detect meaningful departures from

Statisticians spend a lot of time trying to figure out a priori if a study is large enough to detect meaningful departures from a null hypothesis

H 0

H 0


Lecture 6 Review: P-values

Small p-values indicate that you have rejected the null hypothesis

If p < 0.05, this means that you have rejected the null hypothesis with a confidence interval of 95% or a Type I error rate of 0.05

If p > 0.05, you did not reject the null hypothesis at these levels


Lecture 6 Review: Statistical Power

Statistical power is defined as the probability that you will reject the null hypothesis when you should reject it.

If is the Type II error, power = 1 -

The Type I error (test level) does NOT depend on the sample size: you chose it (5%?)

The power depends crucially on the sample size


Sample Size Calculations

You want to test at level (Type I error) the null hypothesis that the mean = 0

• You want power 1 - to detect a change of from the hypothesized mean by the amount or more, i.e., the mean is greater than or the mean is less than -

• There is a formula for this!!


Sample Size Calculations

Look up z and z

Remember what they are?

Find the values in Table 1 which give you readings of 1-and 1-

Required sample size is 2

22

2

zzn


Sample Size Calculations =0.01=0.90=180=600

Look up z =2.58 and z=1.28 (Check this)

= 166

=0.01=0.80=180=600, z=0.84 (Check this)

n = 130: the less power you want, the smaller the sample size

2

22

2

zzn


More on Sample Size Calculations

Most often, sample sizes are done by convention or convenience:

Your professor has used 5 rats/group before successfully

You have time only to interview 50 subjects in total


More on Sample Size Calculations

More often, sample sizes are done by convention or convenience:

In this case, the sample size calculations can be used after a study if you find no statistically significant effect

You can then guess how large a study you would have needed to detect the effect you have just seen but which was not statistically significant


Never Accept a Null Hypothesis

Suppose we use a 95% confidence interval, it includes zero. Why do I say: with 95% confidence, I cannot reject that the population mean is zero.

I never, ever say: I can therefore conclude that the population mean is zero.

Why is this? Are statisticians just weird? (maybe so, but not in this case)


Never Accept a Null Hypothesis: Reason 1

Suppose we use a 95% confidence interval, it includes zero: [-3,6]. Why do I say: with 95% confidence, I cannot reject that the population mean is zero.

Remember the definition of a confidence interval: the chance is 95% that the true population mean is between -3 and 6: hence, the true population mean could be 5, and is not necessarily = 0.



Suppose we use a 95% confidence interval, it includes zero: [-3,6]. Why do I say: with 95% confidence, I cannot reject that the population mean is zero.

Potential for chicanery: if you want to accept the null hypothesis, how can you best insure it?



An example of chicanery: generic drugs

In the pharmaceutical industry, all the expense involves getting a drug approved by the FDA

After a drug goes off-patent, generic drugs can be marketed

The main regulation is that the generic must be shown to be “bioeqiuvalent” to the patent drug



The generic must be shown to be “bioeqiuvalent” to the patent drug

One way would be to run a study and do a statistical test to see whether the drugs have the same effects/actions: the null hypothesis is that the patent and generic are the same

The alternative is that they are not

If the null is rejected, the generic is rejected, and $$$ issues arise



Test to see whether the drugs have the same effects/actions: the null hypothesis is that the patent and generic are the same

If the null is rejected, the generic is rejected, and $$$ issues arise

If you pick a tiny sample size, there is no statistical power to reject the null hypothesis



If you pick a tiny sample size, there is no statistical power to reject the null hypothesis

The FDA is not stupid: they insist that the sample size be large enough that any medically important differences can be detected with 80% (1 - ) statistical power


Never Accept a Null Hypothesis

p-values are not the probability that the null hypothesis is true.

For example, suppose you have a vested interest in not rejecting the null hypothesis.

Small sample sizes have the least power for detecting effects.

Small sample sizes imply large p-values.

Large p-values can be due to a lack of power, or a lack of an effect.


Paired Comparisons: Count you Number of Populations!

The hormone assay data illustrate an important point.

Sometimes, we measure 2 variables on the same individuals

Reference Method and Test Method

There is only 1 population. How do we compare the two variables to see if they have the same mean?


Paired Comparisons: Count you Number of Populations!

There is only 1 population. How do we compare the two variables to see if they have the same mean?

Answer (Ott & Longnecker, Chapter 6.4): do what we did and first compute the difference of the variables and make inference on this difference: now have 1 variable

In making inference, match the number of variables to the number of populations!


Paired Comparisons in SPSS

SPSS has a nice routine way of doing a paired comparison analysis, providing confidence intervals and p-values

“Analyze”

“Compare Means”

“Paired Samples t-test”

Highlight the variables that are paired and select: use “options” to get other than 95% CI


Paired Comparisons in SPSS

Demo using computer comes next


Boxplots and Histograms for Paired data

For paired data, SPSS makes it easy to automatically get confidence intervals: it takes the difference of the paired variables for you

However, for boxplots, qq-plots, etc., you have to do this manually.

Here is how you can define a new variable, called “differen”, in the armspan data for males.


Computing the Difference in Paired Comparisons

Click on “Transform”

Click on “Compute”

New window shows up, in “Target Variable” type in differen

Click on “Type & Label” and type in your label (Height - Armspan in Inches)

click on “Continue”


Computing the Difference in Paired Comparisons

Highlight height and move over by clicking the mover button

In “Numeric Expression”, type in the minus sign -

Highlight armspan and move over

Click on “OK”

You are done!


Selecting Cases in SPSS

“Data”

“Select Cases”

Push button of “If condition is satisfied”

Select “If”

Select “Gender” and move over

Then type = ‘Female’ and “Continue”

“OK” --> all analyses will be on Females


Student’s t-Distribution

In real life, the population standard deviation is never known

We estimate it by the sample standard deviation s

To account for this estimation, we have to make our confidence intervals (make a guess): longer or shorter?

Stump the experts!


Student’s t-distribution

Of course: you have to make the confidence interval longer!

This fact was discovered by W. Gossett, the brewmaster of Guinness in Dublin.

He wrote it up anonymously under the name “Student”, and his discovery is hence called Students t-distribution because he used the letter t in his paper.



Effectively, if you want a (1100% confidence interval, what you do is to replace z (1.645, 1.96, 2.58) by

t(n-1) found in Table 2 of the book.

n-1 is called the degrees of freedom

The increase in length of the confidence interval depends on n.

If n gets larger, does the CI get larger or smaller?



The (1100% CI when was known was

The (1100% CI when is unknown is

You replace

by s and

by t(n-1)

/2X z / n

/2X t (n-1)s / n

/2z



Take 95% confidence, = 0.05

z = 1.96

n = 3, n-1 = 2, t(n-1) = 4.303

n = 10, n-1 = 9, t(n-1) = 2.262

n = 30, n-1 = 29, t(n-1) = 2.045

n = 121, n-1 = 120, t(n-1) = 1.98



Luckily, SPSS is smart.

It automatically uses Student’s t-distribution in constructing confidence intervals and p-values!

So, all the output you will see in SPSS has this correction built in



In the old days, people used the t-test to decide whether the hypothesize value is in the CI.

If your hypothesis is that = 0, then you reject the hypothesis if

You learn nothing from this not available in a CI, but its value is in SPSS

/2 /2

Xt = t (n-1) or < -t (n-1)

s / n


WISH Numerical Illustration

s = 613, Xbar = -180

n = 3, s.e. = 613 / 31/2 = 354, t(n-1) = 4.303, CI is -180 plus and minus 1523, hence the interval is [-1703, 1343]

n = 121, s.e. = 613 / 1211/2 = 59, t(n-1) = 1.98, CI is -180 plus and minus 118, hence the interval is [-298,-62]

Note change in conclusions!


Armspan Data for Males

Outcome is height – armspan in inches

In SPSS, “Analyze”, “Descriptives”, “Explore” will get you to the right analysis

Illustrate how to do this in SPSS



Sample mean = -0.26

Sample standard error = 0.2391

Lower bound of 95% CI = -0.7406

Upper bound of 95% CI = 0.2206

Is there evidence with 95% confidence that armspans for males differ systematically from heights?



Might ask: what about with 90% confidence

Illustrate how to do this in SPSS



Sample mean = -0.26

Sample standard error = 0.2391

Lower bound of 90% CI = -0.6609

Upper bound of 90% CI = 0.1409

Is there evidence with 90% confidence that armspans for males differ systematically from heights?



SPSS will compute the p-value for you as well as confidence intervals.

For paired comparisons, “Analyze”, “Compare Means”, “Paired Sample”.

Highlight the paired variables.

It computes the difference of the first named variable in the list minus the second

Illustration in SPSS



t = -1.087

p-value (significance level) = 0.282

SPSS also automatically does a 95% confidence interval for the population mean difference between heights and armspans

Copyright (c) Bani Mallick1 STAT 651 Lecture 7. Copyright (c) Bani Mallick2 Topics in Lecture #7...

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