Session 1 Probability to confidence intervals By : Allan Chang

Probability Mean The normal distribution Standard Deviation (of measurements) Confidence interval of measurements Population and sample Standard Error (of the mean) Confidence interval of the mean Difference between two means (mean of differences) Standard Error (of the difference) Confidence interval of the difference

Probability Philosophical nature of truth –Aristotle : things can be true or false, but cannot be both –Confucius : everything is in the middle Science and empiricism –Truth cannot be known, only estimated from observations –Things repeatedly observed are more likely to be true –No theory can be accepted unless supported by repeated observations Uncertainties in observations –Multiple influences on any observation –Precisions of measurement –Very unlikely to get the same result every time –Probability provides an expression of confidence in the results –Statistics is the set of tools to summarise the observations

How probability is expressed Convention : a number between 0 and 1. –0 (0%) = absolutely impossible, 1 (100%) = absolute certainty Subjective probability : An expression of opinion. –e.g. the probability of my staying awake in this lecture is 10% Logical probability : mathematical calculations based on assumptions, how most statistical theories are developed –e.g. If the probability of getting heads in tossing a coin is 0.5 getting 2 heads in 2 tosses is 0.5x0.5=0.25, getting 2 tails = 0.25, and getting 1 head and 1 tail = 0.5 Getting at least 1 heads in 2 tosses is (0.5x0.5) = 0.75 Empirical probability : calculated from observations –e.g. if there are 10 women and 20 men in the class, the probability of being male is 20/(10+20) = 0.67 (67%)

Probability Mean The normal distribution Standard Deviation (of measurements) Confidence interval of measurements Population and sample Standard Error (of the mean) Confidence interval of the mean Difference between two means (mean of differences) Standard Error (of the difference) Confidence interval of the difference

Mean Havara : a system of insurance amongst Phoenecian traders Havara -> average -> mean Mean is the centre of all the measurements

Probability Mean The normal distribution Standard Deviation (of measurements) Confidence interval of measurements Population and sample Standard Error (of the mean) Confidence interval of the mean Difference between two means (mean of differences) Standard Error (of the difference) Confidence interval of the difference

Gauss De Moivre

Probability Mean The normal distribution Standard Deviation (of measurements) Confidence interval of measurements Population and sample Standard Error (of the mean) Confidence interval of the mean Difference between two means (mean of differences) Standard Error (of the difference) Confidence interval of the difference

De Moivre Fisher

z p Standard deviate z Probability p Z=0; p=0.5 Z=1.29; p=0.1 Z=1.65; p=0.05 Z=1.96; p=0.025 Z=2.33; p=0.01 Z=2.58; p=0.005 Z=3.09; p=0.001 Fisher

Probability Mean The normal distribution Standard Deviation (of measurements) Confidence interval of measurements Population and sample Standard Error (of the mean) Confidence interval of the mean Difference between two means (mean of differences) Standard Error (of the difference) Confidence interval of the difference

Confidence interval Half of excluded each side e.g. in 95% CI 5% is excluded 2.5% each side (z=1.96) 95% CI = mean +/ SD

Confidence interval of measurements The range any single measurement is likely to be within CI measurement = mean +/- z SD –90% CI = mean +/ SD –95% CI = mean +/ SD –99% CI = mean +/ SD

Probability Mean The normal distribution Standard Deviation (of measurements) Confidence interval of measurements Population and sample Standard Error (of the mean) Confidence interval of the mean Difference between two means (mean of differences) Standard Error (of the difference) Confidence interval of the difference

Population –Measure everyone, the results are true of the population –Very expensive and inconvenient to do, in most cases not possible Sampling –Measure a small but representative portion of the population, and assume that the results are true also of the population as a whole –The results are estimates of the truth, approximations –The degree of approximation (error) is important and needs to be estimated

Probability Mean The normal distribution Standard Deviation (of measurements) Confidence interval of measurements Population and sample Standard Error (of the mean) Confidence interval of the mean Difference between two means (mean of differences) Standard Error (of the difference) Confidence interval of the difference

Standard Error of the mean –Conceptually, the Standard Deviation of all the mean values, if the samples are taken repeatedly

nmeanSDSE mean and SD of the 10 means Standard Error of the mean SE = SD / sqrt(n) Random number mean=0 SD=1 10 samples of 20 each SE each sample = 0.2 – 0.32 SD of the 10 means = 0.2 SE is an estimate of the SD of the means if repeated sampling of the same size occurs

Standard Error of the mean (SE) –A function of Standard Deviation, and sample size –If the sample size is infinite, or the whole population, then SE = 0, as the mean value has no error –If the sample size =1, then the SE is the same as the Standard Deviation –SE mean = SD / sqrt(n)

Sample size SE = SD / SQRT(n) SE as % of SD

Probability Mean The normal distribution Standard Deviation (of measurements) Confidence interval of measurements Population and sample Standard Error (of the mean) Confidence interval of the mean Difference between two means (mean of differences) Standard Error (of the difference) Confidence interval of the difference

Confidence interval of the mean –The range within which the true mean value can be expected –CI mean = mean +/- z SE 90% CI = mean +/ SE 95% CI = mean +/ SE 99% CI = mean +/ SE –CI mean = CI measurement / SQRT(n)

Probability Mean The normal distribution Standard Deviation (of measurements) Confidence interval of measurements Population and sample Standard Error (of the mean) Confidence interval of the mean Difference between two means (mean of differences) Standard Error (of the difference) Confidence interval of the difference

Theory of sampling –Difference between the means of samples in two groups can be considered as the mean difference between these two groups in the population –e.g. The difference in the mean weights between 200 men and 200 women represents the mean difference in weight between men and women in the population –The problem is the Standard Error and the confidence interval

Probability Mean The normal distribution Standard Deviation (of measurements) Confidence interval of measurements Population and sample Standard Error (of the mean) Confidence interval of the mean Difference between two means (mean of differences) Standard Error (of the difference) Confidence interval of the difference

R.A. Fisher devised the Analysis of Variance between groups –Total variations in measurements are partitioned That attributable to groups difference That attributable to individual differences (residual, error)

From the Analysis of variance –Individual or random variation is partitioned –What is left are group variance and difference –From this Further partition to Standard Errors for each group What is left then is the variation or error of the difference From this, the Standard Error of the difference is derived SE difference is a function of the sample size and SD of the two groups, and the difference in the means between the two groups

Probability Mean The normal distribution Standard Deviation (of measurements) Confidence interval of measurements Population and sample Standard Error (of the mean) Confidence interval of the mean Difference between two means (mean of differences) Standard Error (of the difference) Confidence interval of the difference

–The range within which the difference can be expected –CI difference = mean +/- z SE difference 90% CI = difference +/ SE difference 95% CI = difference +/ SE difference 99% CI = difference +/ SE difference

Interpretation of CI difference CI represent the range within which the “true value” is likely to be –CI that does not traverse the null value means that the difference is not null, or that the difference is statistically significant –CI that are inside of a meaningless difference (tolerance) means that the difference is not of practical consequence, and can be considered to be equivalent

An example of CI We do two studies –Study 1, the difference in weight between women and men –Study 2, the difference in weight between those named Jim and all those named Bill –We think that a difference of less than 1Kg is probably not big enough to worry about

Difference in weight (Kg) men – women n = 20 men – women n = 100 Jim – Bill n = 200 Jim – Bill n = 1000 Tolerance +/- 1Kg

Terms you need to know Measurements –Mean, Standard Deviation, confidence interval Means –Mean, Standard Error of the mean, confidence interval Difference –Difference between means, mean difference, Standard Error of the difference, confidence interval, significant difference, not significantly different, equivalence