Sampling & Estimation

Normal Distribution

Normal Sample

Binomial Distribution

Estimation

Sampling

Sampling of the Mean

The more observations the better!

Surprice!!!!!

Sampling of the Variance

Sampling of the proportion

How accurate are these estimates?

Can we use that to report the uncertainty

in a clever way?

Rule of

A random variable is very seldom more than two standard deviations away from the expected value.

A random variable is very seldom more than two standard deviations away from the expected value.

… Ehh, we don’t know that one!

Confidence Interval for the Mean when the variance is not know

Confidence intervals for the variance

It looks like …..

A 95% approximate interval for a proportion

Assume normality

BUT WHAT IF THIS INTERVAL

CONTAINS 0 OR 1?This would be possible if n is small, if p is nearly zero or if p is nearly one.

Log-Transformation

Believe me!Assume normality

Use the expontial transformation, and write

But what if the interval contains

one?

This could happen if n is relatively small and p is nearly one.

Logit-transformation

and it also looks like log(1-p), for p approx one.

Looks like the

log-transformation, for p small

To go the other way

The function logit(p) The function expit(p)

Logit-transformation

Assume normality

To get a 95% CI for p, we use the expit-transformation

Now we are happy!

Why didn’t I just tell you about the logit-transformation in the first place?Because, when comparing proportions (risks), you may consider

To get 95% CI here, you’ll need all three approaches.

How to calculate CI’s in SPSS

• It is easy (sort of) in the case of normally distributed variables

• More or less impossible in case of binomial (Use Excel)

Assume we have a dataset with a variable called: Alcohol

Hmmmm

Choose

• Analyze

• General Linear Model

• Univariate

Choose

• Analyze

• General Linear Model

• Univariate

• Drag the variable Alcohol into Dependent Variable

• Click Options

• Choose Parameter estimates

• Drag the variable Alcohol into Dependent Variable

• Click Options

• Choose Parameter estimates

… And now we get