Introduction to Statistics Dr Linda Morgan Clinical Chemistry Division School of Clinical Laboratory Sciences

Outline Types of data Descriptive statistics Estimates and confidence intervals Hypothesis testing Comparing groups Relation between variables Statistical aspects of study design Pitfalls

Types of data Categorical data –Ordered categorical data Numerical data –Discrete –Continuous

Descriptive statistics Categorical variables Graphical representation – bar diagram Numbers and proportions in each category

Descriptive statistics Continuous variables Distributions –Gaussian –Lognormal –Non-parametric Central tendency –Mean –Median Scatter –Standard deviation –Range –Interquartile range

Gaussian (normal) distribution

Central tendency Mean =  x n Scatter Variance =  (x-mean) 2 n –1 Standard deviation =  variance Gaussian (normal) distribution

Lognormal distribution

Mean =  log x n Geometric mean = antilog of mean (10 mean ) Median –Rank data in order –Median = (n+1) / 2 th observation

Variability Variance =  (x-mean) 2 n –1 Standard deviation =  variance Range Interquartile range

Variability of Sample Mean The sample mean is an estimate of the population mean The standard error of the mean describes the distribution of the sample mean Estimated SEM = SD/  n The distribution of the sample mean is Normal providing n is large

Standard error of the difference between two means SEM = SD/  n Variance of the mean = SD 2 /n Variance of the difference between two sample means = sum of the variances of the two means = (SD 2 /n) 1 + (SD 2 /n) 2 SE of difference between means =  [ (SD 2 /n) 1 + (SD 2 /n) 2 ]

Variability of a sample proportion Assume Normal distribution when np and n(1-p) are > 5 SE of a Binomial proportion =  (pq/n) where q = 1-p

Standard error of the difference between two proportions SE (p 1 – p 2 ) =  [variance (p 1 ) + variance (p 2 ) ] =  [ ( p 1 q 1 /n 1 ) + ( p 2 q 2 /n 2 ) ]

Confidence intervals of means 95% ci for the mean = Sample mean  1.96 SEM 95% ci for difference between 2 means = (mean 1 – mean 2 )  1.96 SE of difference

Confidence intervals of proportions 95% ci for proportion = p  1.96  (pq/n) 95% ci for difference between two proportions = (p 1 – p 2 )  1.96 x SE (p 1 – p 2 )

Hypothesis testing The null hypothesis The alternative hypothesis What is a P value?

Comparing 2 groups of continuous data Normal distribution: paired or unpaired t test Non-Normal distribution: transform data OR Mann-Whitney-Wilcoxon test

Paired t test We wish to compare the fasting blood cholesterol levels in 10 subjects before and after treatment with a new drug. What is the null hypothesis?

Paired t test SubjectFasting cholesterolD NumberPredrugPostdrug

Paired t test Calculate the mean and SEM of D The null hypothesis is that D = 0 The test statistic t = mean(d) – 0 SEM (d)

Paired t test Mean = 0.62 SEM = t = Degrees of freedom = n - 1 = 9 From tables of t, 2-tailed probability (P) is between 0.1 and 0.2 How would you interpret this?

Comparing 2 groups of categorical data In a study of the effect of smoking on the risk of developing ischaemic heart disease, 250 men with IHD and 250 age-matched healthy controls were asked about their current smoking habits. What is the null hypothesis?

Results 70 of the 250 patients were smokers 30 of the healthy controls were smokers SmokerNon- smoker Total IHD Control Total

SmokerNon-smokerTotal IHD Control Total Calculate expected values, E, for each cell

Calculate (observed – expected) value, D SmokerNon-smokerTotal IHD 70 – 50 = –200= -20 Control 30-50= = 20 Total

Calculate D 2 /E SmokerNon-smokerTotal IHD 400/50= 8 400/200= 2 Control 400/50= 8 400/200= 2 Total

Calculate the sum of D 2 /E = 20 This is the test statistic, chi squared Compare with tables of chi squared with (r-1)(c-1) degrees of freedom In this case, chi squared with 1 df has a P value of < How do you interpret this?

Statistical analysis using computer software SPSS as an example

Planning Experimental design Suitable controls Database design

Statistical power The power of a study to detect an effect depends on: –The size of the effect –The sample size The probability of failing to detect an effect where one exists is called  The power of a study is 100(1-  )% Wide confidence intervals indicate low statistical power

Statistical power The necessary sample size to detect the effect of interest should be calculated in advance Pilot data are usually required for these calculations

Statistical power - example 30% of the population are carriers of a genetic variant. You wish to test whether this variant increases the risk of Alzheimers Disease. For P < 0.05, and 80% power, number of controls and cases required: Control carriersCase carriersSample size 30%50% %40% %35%1400

Multiple testing Number ofProbability of Testsfalse positive Bonferroni correction: Divide 0.05 by the number of tests to provide the required P value for hypothesis testing at the conventional level of statistical significance

Data trawling Decide in advance which statistical tests are to be performed Post hoc testing of subgroups should be viewed with caution Multiple correlations should be avoided

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