P Values - part 2Samples & Populations

Robin Beaumont11/02/2012

With much help from

Professor Chris Wilds material University of Auckland

Aspects of the P value

P Valuesampling

probability

statistic Rule

Resume

P value = P(observed summary value + those more extreme |population value = x)

A P value is a conditional probability considering a range of outcomes

Sample value

Hypothesised population value

The Population

Ever constant at least for your study!

= Parameter

Sample estimate = statistic

P value = P(observed summary value + those more extreme |population value = x)

One sample

Many thanks Professor Chris Wilds at the University of Auckland for the use of your material

Size matters – single samples

Many thanks Professor Chris Wilds at the University of Auckland for the use of your material

Size matters – multiple samples

Many thanks Professor Chris Wilds at the University of Auckland for the use of your material

We only have a rippled mirror

Many thanks Professor Chris Wilds at the University of Auckland for the use of your material

Standard deviation - individual level

= measure of variability within sample

'Standard Normal distribution'

Total Area = 1

0 1= SD value

68% 95%

2

Area:

Between + and - three standard deviations from the mean = 99.7% of area Therefore only 0.3% of area(scores) are more than 3 standard deviations ('units') away.

-

But does not take into account sample size

= t distribution

Defined by sample size aspect ~ df

Remember the previous tutorial

Sampling level -‘accuracy’ of estimate

From: http://onlinestatbook.com/stat_sim/sampling_dist/index.html

= 5/√5 = 2.236

SEM = 5/√25 = 1

We can predict the accuracy of your estimate (mean)

by just using the SEM formula.

From a single sample

Talking about means here

Standard deviation

sample size

Example - Bradford Hill, (Bradford Hill, 1950 p.92)• mean systolic blood pressure for 566 males around Glasgow = 128.8

mm. Standard deviation =13.05 • Determine the ‘precision’ of this mean.

• SEM formula (i.e 13.5/ √566) =0.5674• “We may conclude that our observed mean may differ from the true

mean by as much as ± 1.134 (.5674 x 2) but not more than that in around 95% of samples.” page 93. [edited]

All possible values of

mean

125 126 127 128 129 130 131x

We may conclude that our observed mean may differ from the true mean by as much as ± 1.134 (.5674 x 2) but not more than that in around 95% of samples.”That is within the range of 127.665 to 129.93

125 126 127 128 129 130 131x

The range is simply the probability of the mean of the sample being within this interval

P value = P(observed summary value + those more extreme |population value = x)

P value of near 0.05 = P(getting a mean value of a sample of 129.93 or one more extreme in a sample of

566 males in Glasgow |population mean = 128.8 mmHg )

in R to find P value for the t value 2*pt(-1.99, df=566) = 0.047

Variation what have we ignored!

Sampling summary• The SEM formula allows us to:• predict the accuracy of your estimate

( i.e. the mean value of our sample) • From our single sample• Assumes we have a Random sample

Aspects of the P value

P Valuesampling

probability

statistic Rule