Measurement and Significant Figures

How confident are we in the estimation of mean/proportion we have calculated?

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Measures of precision:Standard error of mean, SEM Standard error of proportion, SE(p)

Confidence interval for mean Confidence interval for proportion

Standard error of mean, SEM

Number of patientsStandard deviation, SDSEM is smaller (estimate is more precise):the larger is N (number of patients)the smaller is SD (dispersion of data)

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95% confidence interval for mean, 95% CITogether with SEM, 95% CI is also the measure of precision

Unlike SEM, 95% CI also estimates accuracy of the resultie. 95% is accurate that interval includes true (population) mean)

95% confidence interval for meanIf we draw a 100 samples from our population we would find the true population value within 95% confidence interval in 95 samples. 9

20 samplesCritical values for 90%, 95% and 99% level of confidence90% CI => mean 1.65 SEM95% CI => mean 1.96 SEM 99% CI => mean 2.58 SEM Level of Confidence - Critical Value 0.75, or 75% 1.15 0.80, or 80% 1.28 0.85, or 85% 1.44 0.90, or 90% 1.65 0.95, or 95% 1.96 0.98, or 98% 2.33 0.99, or 99% 2.58Example 1The average systolic BP before treatment in study A, of a group of 100 hypertensive patients, was 170 mmHg. After treatment with the new drug the mean BP dropped by 20 mmHg.If the 95% CI is 1525, this means:11we can be 95% confident that the trueeffect of treatment is to lower the BP by 1525 mmHg.Example 2In study B 50 patients were treated with the same drug, also reducing their mean BP by 20 mmHg, but with a wider 95% CI of -5 to +45. This CI includes zero (no change). This means:12there is more than a 5% chance that there was no true change in BP, and that the drug was actuallyineffective..Example 3 Meta analysis

Fig. Plot of 5 studies of a new antihypertensive drug.Which study showed the greatest change?Did all the studies show change in favour of the intervention?Were the changes statistically significant?

Results of studies A and B above are shown by the top two lines, i.e. 20 mmHg, 95% CI 1525 for study A and 20 mmHg, 95% CI -5 to +45 for study B.

13Watch out for...

The size of a CI is related to the sample size of the study. Larger studies usually have a narrower CI.14ProportionStandard error of proportion, SE(p) SE(p) = (p(1 p)/n)

Confidence interval for proportion

The standard deviation describes the variability of a sample; The standard error of the mean (SEM) does not describe the sample but describes the uncertainty of how the sample mean represents the population mean.SDCIStandard deviation tells us about the variability (spread) in a sample.The CI tells us the range in which the true value (the mean if the sample were infinitely large) is likely to be.Krebs NF, Westcott JE, Culbertson DL et. al. Comparison of complementary feeding strategies to meet zinc requirements of older breastfed infants. Am J Clin Nutr. 2012; 96:30-35 Mean (SEM) total absorbed zinc amounts were 0.80 0.08, 0.71 0.09, and 0.52 0.05 mg/d for the: meat, iron-and-zinc-fortified infant cereal, and whole-grain, iron-only-fortified infant cereal groups of infants.

SEMCIMeat Fe&Zn Fe

Meat Fe&Zn Fe

TRUE or FALSEWhat does a small standard error tell us about the sample estimate of the mean?That it is highly variableThat the population standard deviation may be smallThat the sample size is probably smallThat it is imprecise

TRUE or FALSEWhat will tend to make the standard error larger?A small varianceA large standard deviationImprecise dataInaccurate data

20Statistical inference:Hypothesis testing21Statistics: Learning from Samples about Populations

Inference 1: Confidence IntervalsWhat does the 95% CI really mean?

Inference 2: Hypothesis TestsWhat does a p-value really mean?When to use which test?

Statistical Inference: Brief Overview 22In epidemiological studies: Is there a relationship between a variable of interest and an outcome of interest?Ie. smoking and lung cancerStress and thyroid cancer

In clinical trails: Is experimental therapy more effective than standard therapy or placebo?

Examples of hypothesis testing in medical research

Hypothesis testing = testing of statistical hypothesis24Statistical hypothesisStatements about population parameter values.

Null hypothesis (H0) says a parameter is unchanged from a default, pre-specified value; andAlternative hypothesis (H1) says parameter has a value incompatible with H0

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Make appropriate statistical hypotheses: Assumption: Mean cholesterol in hypertensive men is equal to mean cholesterol in male general population (20-74 years old). In the 20-74 year old male population the mean serum cholesterol is 211 mg/ml with a standard deviation of 46 mg/ml

Example: Hypertension and CholesterolNull hypothesis => no difference between treatmentsH0: hypertensive = general populationH0: hypertensive = 211 mg/ml = population mean of serum cholesterol Mean cholesterol for hypertensive men = mean for general male population

Alternative hypothesisHA: hypertensive general populationHA: hypertensive 211 mg/ml

Example: Hypertension and CholesterolNull and alternative hypothesis28

Two-sided testsOne-sided testsHow to choose one or the other?29

Steps in Hypothesis TestsAssume H0 is true i.e. believe results are a matter of chance

Quantify how far away are data from being consistent with H0 by evaluating quantity called a test statistic

Assess probability of results at least this extreme - call this the p-value of the test

Reject H0 (believe H1) if this p-value is small or keep H0 (do not believe H1) otherwise

Interpretation of P-value (0.05)P>=0.05Significant difference between the treatmentsNull hypothesis is rejected, alternative is acceptedP