Hypothesis - Biostatistics
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Hypothesis - Biostatistics

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Understanding Hypothesis in Biostatistics for more details check our website at

Understanding Hypothesis in Biostatistics for more details check our website at
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Hypothesis - Biostatistics Hypothesis - Biostatistics Presentation Transcript

  • Biostatistics Lecture 8
  • Lecture 7 Review– Using confidence intervals and p-values to interpret the results of statistical analyses • • • Null hypothesis P-value Interpretation of confidence intervals & p- values
  • Null hypothesis • A null hypothesis is one that proposes there is no difference in outcomes • We commonly design research to disprove a null hypothesis
  • P-value:- comparing two groups What is the probability (P-value) of finding the observed difference How likely is it we would see a difference this big IFIF The null hypothesis is true? There was NO real difference between the populations?
  • Interpretation of p-values 1! Weak evidence against the null hypothesis0.1! Increasing evidence against the null hypothesis with decreasing P-value 0.01! 0.001! Strong evidence against the null hypothesis 0.0001! P-value!
  • Objective To assess the effect of combined hormone replacement therapy on health related quality of life. Design Randomised placebo controlled double blind trial. (HRT) Table 3 EuroQoL Visual Analogue Scores (EQ-VAS) by treatment group. Figures are means (SE) one year (95% Combined HRT (n=1043*) Placebo (n=1087*) Adjusted difference at CI) P-value EQ-VAS 77.9 (0.5) 78.5 (0.4) -0.59 (-1.66 to 0.47) 0.28
  • Five trials of drugs to reduce serum cholesterol A reduction of 0.5 mmol/L or more corresponds to a clinically important effect of the drug Trial Drug Cost No. of patients per group Observed difference in mean cholesterol (mmol/L) s.e. of difference (mmol/L) 95% CI for population difference in mean cholesterol P-value 1 A Cheap 30 -1.00 1.00 -2.96 to 0.96 0.32 2 A Cheap 3000 -1.00 0.10 -1.20 to -0.80 <0.001 3 B Cheap 40 -0.50 0.83 -2.13 to 1.13 0.55 4 B Cheap 4000 -0.05 0.083 -0.21 to 0.11 0.55 5 C Expensive 5000 -0.125 0.05 -0.22 to -0.03 0.012
  • Lecture 8 – Proportions and intervals Binary variables (RECAP) confidence • • Single proportion – Standard error, confidence interval • Incidence & prevalence • Difference in two proportions – Standard error, confidence interval
  • Categorical variables - Binary Binary variable – two categories only (also termed – dichotomous variable) Examples:-  Outcome – Diseased or Healthy; Alive or Dead…  Exposure - Male or Female; Smoker or non- smoker; Treatment or control group….
  • Inference Proportion of population diseased – π?? Proportion of sample diseased, p=d/n Number of subjects who do experience outcome (diseased) = d Number of subjects who do not experience outcome (healthy) = h Total number in sample = n = h + d
  • Inference - example Proportion of population with vivax malaria - π Proportion of sample with vivax p = d/n = 15/100 = 0.15 (15%) malaria, Number of sample with vivax malaria = d = 15 Number of sample without vivax malaria = h = 85 Total number in sample = n = 15 + 85 = 100
  • Single proportion - Inference • Obtain a sample estimate, p, of the population proportion, π • REMEMBER different samples would give different estimates of π (e.g. sample 1 p1, sample 2 p2,…) • Derive: – Standard error – Confidence interval
  • Standard error & confidence interval of a single proportion • Standard error (SE) for single proportion:- (from the Binomial distribution) π (1−π ) p(1− p) s.e.( p) = ~ n n • 95% CI for single proportion:- (approximate method based on the normal distribution) – – Lower limit = p - 1.96×s.e.(p) Upper limit = p + 1.96×s.e.(p)
  • Standard error & confidence interval a single proportion – malaria exampleof • • Estimated proportion of vivax Standard error of p malaria (p) = 15/100 = 0.15 p(1− p) 0.15(1−0.15) s e ( p). . = = 0.036= n 100 • 95% Confidence interval for population proportion (π) – – Lower limit = p - 1.96×s.e.(p) = 0.15 – 1.96×0.036 = 0.079 Upper limit = p + 1.96×s.e.(p) = 0.15 + 1.96×0.036 = 0.221 Interpretation.. “We are 95% confident, the population proportion of people vivax malaria is between 0.079 and 0.221 (or between 7.9% and 22.1%)” with
  • Definition of a confidence REMEMBER….. interval If we were to draw several independent, random samples (of equal size) from the sample population and calculate 95% confidence intervals for each of them, 0. 4 0.3 5 0. 3 Populatio n0.2 5 then on average 19 out of every 20 (95%) such confidence intervals would contain the true population proportion (π), and one of every 20 0. 2 0.1 5 0. 1 (5%) would not. 0.0 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Sample Sampleproportionand95%CI proportion = 0.16 (16%)
  • WARNING…. Confidence Interval of a single proportion The normal approximation method breaks down 1) 2) if: Sample Sample size (n) is small proportion (p) is close to 0 or 1 Require: np ≥ 10 or n(1-p) ≥ 10 Stata lets you calculate an ‘exact’ CI
  • Confidence Interval for a single proportion in Stata • cii 100 15 • • • • -- Binomial Exact -- [95% Conf. Interval]Variable | Obs Mean Std. Err. -------------+-------------------------------------------------------------- - | 100 .15 .0357071 .0864544 .2353075 • cii 100 1 • • • • -- Binomial Exact -- [95% Conf. Interval]Variable | Obs Mean Std. Err. -------------+-------------------------------------------------------------- - | 100 .01 .0099499 .0002531 .0544594
  • Interpretation of proportions: Incidence versus Prevalence
  • Prevalence Proportion of people in a defined population that have a given disease at a specified point in time • Prevalence = no. of people with the disease at particular point in time no. of people in the population at a particular point in time Examples:- • Prevalence living in the Prevalence Thailand. Prevalence of chronic pain among people aged 25+ years and Grampian region, UK. of typhoid among villagers living in Tak province,• • of diagnosed asthma in individuals aged 15 to 50 years, registered with a particular general practice in Carlton.
  • Incidence risk (Cumulative incidence) Proportion of new cases in a disease free population in a given time period • Incidence risk = no. of new cases of disease in a given time period no. of people disease-free at beginning of time period Examples:- • Incidence risk of death in five years following diagnosis with prostate cancer Incidence risk of breast cancer over 10 years of follow-up in women 40-69 years of age and free from breast cancer in 1990 •
  • Incidence rate (NOT a proportion) Number of new cases in a disease free population per person per unit time • that occur Incidence rate = no. of new cases of disease total person-years of observation Examples:- • Incidence rate of all-cause mortality of men in the Melbourne Collaborative Cohort Study = 9.0 per 1000 men per year ‘9 out of every 1000 men die each year’ (
  • Comparing two proportions
  • Comparing two proportions 2×2 table • • • Proportion Proportion Proportion of all subjects experiencing outcome, p = d/n in exposed group, p1 = d1/n1 in unexposed group, p0 = d0/n0 Be alert (not alarmed): watch for transposing the table and swapping columns or rows With outcome (diseased) Without outcome (disease-free) Total Exposed (group 1) d1 h1 n1 Unexposed (group 0) d0 h0 n0 Total d h n
  • Comparing two proportions Example:- TBM trial (Thwaites GE et al 2004) Adults with tuberculous meningitis randomly allocated into treatment groups: 2 1. 2. Dexamethasone Placebo Outcome measure: Death during nine months following start of treatment. Research question: Can treatment with dexamethasone reduce the risk of death adults with tuberculous meningitis? among
  • Comparing two proportions Example – TBM trial Death during 9 months post start of treatment Treatment group Yes No Total Dexamethasone (group 1) 87 187 274 Placebo (group 0) 112 159 271 Total 199 346 545
  • Difference in two population proportions, π1-π0 Estimate of difference in population proportions = p1 – p0 Example:- TBM trial Dexamethasone p1 = d1/n1 = 87/274 = 0.318 Placebo p0 = d0/n0 = 112/271 = 0.413 p1 – p0 = 0.318 – 0.413 = -0.095 (or -9.5%)
  • Difference in two proportions - Inference • Obtain a sample estimate, p1-p0, of the difference in population proportions, π1Dπ0 • REMEMBER different samples of π1Dπ0 (e.g. sample 1 p11-p10, would give different estimates sample 2 p21-p20,…) • Derive: – Standard error of difference in sample proportions – Confidence interval of difference in population proportions
  • Standard error & confidence interval for difference between two proportions • Standard error (SE) for difference between sample proportions:- [s.e.( p )]2 +[s.e.( p )]2 s.e.( p ) =− p 1 0 1 0 • 95% CI for difference between population Lower limit = (p1-p0) - 1.96×s.e.(p1-p0) Upper limit = (p1-p0) + 1.96×s.e.(p1-p0) proportions:-
  • Standard error & confidence interval for difference between two proportions Example:- TBM trial Estimate of difference in population proportions = p1-p0 = -0.095 s.e.(p1-p0) = 0.041 95% CI for difference in population proportions (π1-π0): -0.095 ± 1.96×0.041 -0.175 up to -0.015 OR -17.5% up to -1.5% Interpretation:- “We are 95% confident, that the difference in population proportions is between -17.5% (dexamethasone reduces the proportion of deaths by a large amount) and -1.5% (dexamethasone marginally reduces the proportion of deaths)”.
  • Comparing proportions using csi 87 112 187 159 Stata | Exposed Unexposed | Total -----------------+------------------------+------------ Cases | Noncases | 87 187 112 159 | | 199 346 -----------------+------------------------+------------ Total | | | | | 274 271 | | | | | 545 Risk .3175182 .4132841 .3651376 Point estimate [95% Conf. Interval] |------------------------+------------------------ Risk difference Risk ratio Prev. frac. ex. Prev. frac. pop | | | | -.0957659 .7682808 .2317192 .1164974 | | | | -.1762352 -.0152966 .6139856 .0386495 .9613505 .3860144 +------------------------------------------------- chi2(1) = 5.39 Pr>chi2 = 0.0202 Remember the warning about how the table is presented -Stata requires presentation with outcome by rows and exposure by columns Results are close to those obtained by hand
  • Difference between two proportions:- Risk difference Example:- TBM trial Outcome measure: Death during nine months treatment. following start of Dexamethasone p1 (incidence risk) = d1/n1 = 87/274 = 0.318 Placebo p0 (incidence risk) = d0/n0 = 112/271 = 0.413 p1 – p0 (risk difference) = 0.318 – 0.413 = -0.095 (or -9.5%)
  • Lecture 8 – Objectives • Define binary variables, prevalence and incidence risk • Calculate and interpret a proportion and 95% confidence interval for the population proportion • Calculate and interpret the difference in sample proportions and 95% confidence interval for difference in population proportions
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