HRP223 2008

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HRP 223 - 2008

Topic 8 – Analysis of Means

HRP223 2008Normally Distributed Not Normally Distributed

One sample vs. population One sample t-test Wilcoxon Signed Rank

Two paired samples Paired t-test Difference then Signed Rank

Two unpaired samples T-test Wilcoxon Rank-sum

Three or more unpaired samples ANOVA Kruskal-Wallis

Three or more paired samples Mixed effects Transform then mixed model

Normally Distributed Not Normally Distributed

One sample vs. population Describe > Distribution Describe > Distribution

Two paired samples Analyze >ANOVA>t-test Describe > Distribution

Two unpaired samples Analyze >ANOVA>t-test Describe > Distribution

Three or more unpaired samples Analyze >ANOVA>Linear Analyze >ANOVA>Nonpar.

Three or more paired samples Analyze >ANOVA>Mixed

One Categorical Predictor

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Multiple Categorical Predictors

Unpaired samples– ANOVA

Paired samples– Mixed Effects Models

If data is not normally distributed– There are spcialized statistics (Friedman’s test for 2

predictors).– Try to transform into normally distributed.

Mean vs. Expected BMI

It would be nice to see the actual Excel file.

Adds a link to source

Adds a link to source and runs import wizard

This gives instant access to the current state of the spreadsheet

but it is bugged if you mix character and numeric data.

HRP223 2008

Take a Look at the Data

Prior to analysis, do all 3 plots.

Histograms and box plots show

outliers and bimodal data but are not ideal for

assessing normality.

The formal tests for normality are not great. They will not

find problems with small samples and will declare

problems with large samples.

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...1. 2. 3.

4. 5.1. normal distribution 2. skewed-to-the-right distribution3. skewed-to-the-left distribution4. heavy-tailed distribution5. light-tailed distribution

Image from: Statistics I: Introduction to ANOVA, Regression, and Logistic Regression: Course Notes. SAS Press 2008.

HRP223 2008

Inference 101

You only have one sample but you want to make inferences to the world.

Given what you see in this sample, you can guess what the distribution of samples looks like around the null distribution.

4.26

Freq

uenc

y

1 2 3 4 5 6 7

05

1015

4.06

Freq

uenc

y

1 2 3 4 5 6 7

02

46

812

4.13

Freq

uenc

y

1 2 3 4 5 6 7

05

1015

3.8

Freq

uenc

y

1 2 3 4 5 6 7

05

1015

3.84

Freq

uenc

y

1 2 3 4 5 6 7

05

1015

3.9

Freq

uenc

y

1 2 3 4 5 6 7

02

46

810

3.78

Freq

uenc

y

1 2 3 4 5 6 7

02

46

810

4.13

Freq

uenc

y

1 2 3 4 5 6 7

05

1015

4.01

Freq

uenc

y

1 2 3 4 5 6 7

02

46

812

If the population you are sampling from has a mean of 4,

you will not observe a score of 4.

How do you compare this sample

vs. another with a mean of 5?Make a histogram of the means

.75/sqrt(1) = .75

.75/sqrt(5) = .34

.75/sqrt(25) = .15

1000 samples of size 1

Mean: 4.04 SD: 0.74

Freq

uenc

y

1 2 3 4 5 6 7

040

8012

0

1000 samples of size 5

Mean: 3.99 SD: 0.34

Freq

uenc

y

1 2 3 4 5 6 7

050

150

250

1000 samples of size 25

Mean: 4 SD: 0.15

Freq

uenc

y

1 2 3 4 5 6 7

010

030

0

Dis

tribu

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he M

eans

HRP223 2008

Precision Think of the “+/- something” imprecision in the estimates of

the political polls. You typically end up saying you are 95% sure you chose an

interval that has the true value inside the range bracketed by the confidence limits (CLs). Either the population value is or is not in the interval between the lower and upper confidence limit, and if you repeated the process on many samples, 95% of such intervals would include the population value.

The 99% CI is wider (more accurate) than the more precise 95% CL.

Confidence Intervals from 10 Samples

Axis with units showing your outcome

You want to set the width of the interval so that in 95% of the

experiments, the confidence interval includes the true value.

In theory, you tweak the interval and increase or decrease the width.

The unobservable truth

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Benefits of CLs You have information about the estimate's precision. The width of the CI tells you about the degree of

random error which is set by the confidence interval. Wide intervals indicate poor precision. Plausible

values could be across a broad range.

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Estimation vs. Hypothesis Testing

P-value < .05 corresponds to a 95% CL that does not include the null hypothesis value.

CLs show uncertainty, or lack of precision, in the estimate of interest and thus convey more useful information than the p-value.

CLs vs. p-valuesnull value

Upper CL

LowerCL

Confidenceinterval

P > .05 and the null value is inside of the confidence limits (CLs)

null value

Upper CL

LowerCL

Confidenceinterval

P < .05 and the null value is not inside of the confidence

limits

0 difference between groups or odds ratio of 1

null value

zone of clinical indifference

Not statistically significant and not clinically interesting

Not statistically significant, possibly clinically interesting

Statistically significant but not clinically interesting

Statistically significant and clinically interesting

HRP223 2008

Compare Two Teachers

Import the data Describe the data

– Assign the method as a classification variable Do an unpaired T-test

Do a one-way ANOVA with the predictor having only two levels

SS total is the sum of the distance between each point and the overall

mean line squared.

SS error is the sum of the total squared distances between each point and the group mean lines.

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Psoriasis Scores are arbitrary numbers 0 = < 0%

response, 5 = 26-50% response, etc.

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If and only if you work on a fast machine!

HRP223 2008

Weight Gain

Postpartum Depression

After the Formats

Dementia

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Wide to Long You may have noticed that the data for these

analyses are all set up as long, skinny files where there is a record for every observation on a patient. Some people store data as wide records with many variables with a single record for each person.

To convert from wide to long:– Do data step processing with arrays. – Use the Transpose option on the Data menu.– Combine proc transpose and data step code.– Use a macro I wrote. (It is brand new, so check it.)

Tolerance.sas7bdat is dataset from the book

Save a copy of the macro in a file after the fill in the blanks are done.

The stuff in the blah.sas file:

The stuff in the macro file:

tol: all variables starting with the letters tol

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Narrow to Wide

Of course you can transpose back to wide from narrow.

If you download the keyboard macros today you will see that proc transpose now gives you a code template.