Proportions and Confidence Intervals in Biostatistics
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The document presents a detailed analysis of biostatistical methods related to proportions and confidence intervals, including calculations and interpretations for sample proportions and differences in proportions. It discusses the 95% confidence intervals for population proportions and various measures of association such as risk difference, risk ratio, and odds ratio using a tuberculosis treatment trial as an example. Additionally, it covers statistical inference techniques, confidence interval calculations for ratios, and p-value tests for comparing exposure groups.
![Using Stata
csi 87 112 187 159
| 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 .9613505
.0386495 .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](https://image.slidesharecdn.com/lecture92012-140606010944-phpapp01/85/Proportions-and-Confidence-Intervals-in-Biostatistics-22-320.jpg)
![Using Stata
. csi 87 112 187 159, or
| 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
Odds ratio
|
|
|
|
|
-.0957659
.7682808
.2317192
.1164974
.6604756
|
|
|
|
|
-.1762352
.6139856
.0386495
-.0152966
.9613505
.3860144
.4652544 .937623 (Cornfield)
+-------------------------------------------------
chi2(1) = 5.39 Pr>chi2 = 0.0202
For OR, by default Stata uses Cornfield’s formula for se. You can request
the Woolf formula as csi 87 112 187 159, or woolf](https://image.slidesharecdn.com/lecture92012-140606010944-phpapp01/85/Proportions-and-Confidence-Intervals-in-Biostatistics-26-320.jpg)
![Using Stata – p-value calculated using Chi-squared test
. csi 87 112 187 159, or
| 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
Odds ratio
|
|
|
|
|
-.0957659
.7682808
.2317192
.1164974
.6604756
|
|
|
|
|
-.1762352
.6139856
.0386495
-.0152966
.9613505
.3860144
.4652544 .937623 (Cornfield)
+-------------------------------------------------
chi2(1) = 5.39 Pr>chi2 = 0.0202
For OR, by default Stata uses Cornfield’s formula for se. You can request
the Woolf formula as csi 87 112 187 159, or woolf](https://image.slidesharecdn.com/lecture92012-140606010944-phpapp01/85/Proportions-and-Confidence-Intervals-in-Biostatistics-33-320.jpg)
Proportions and Confidence Intervals in Biostatistics
- 2. Lecture 8 Review – Proportions and confidence intervals • Calculation and interpretation of: – sample proportion – 95% confidence interval for population proportion • Calculation and interpretation of: – – difference in sample proportions 95% confidence interval for difference proportions in population
- 3. Single proportion – Inference • • Estimated proportion of vivax malaria (p) = 15/100 = 0.15 Standard error of p p(1− p) 0.15(1−0.15) s e ( p ). . = = = 0.036 n 100 • 95% Confidence interval for π (population proportion) – – Lower limit = p - 1.96×s.e.(p) = 0.079 Upper limit = p + 1.96×s.e.(p) = 0.221 Interpretation.. “We are 95% confident, the population proportion (π) of people with vivax malaria is between 0.079 and 0.221 (or between 7.9% and 22.1%)”
- 4. 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 With outcome (diseased) Without outcome (disease-free) Total Exposed (group 1) d1 h1 n1 Unexposed (group 0) d0 h0 n0 Total d h n
- 5. Comparing two proportions Example – TBM trial Death during 9 months post start of treatment Treatment group Yes No Total Dexamethasone (group 1) 87 (p1=0.318) 187 274 Placebo (group 0) 112 (p0=0.413) 159 271 Total 199 346 545
- 6. Comparing two proportions - Inference 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)”.
- 7. Comparing two proportions (absolute difference):- 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%)
- 8. Lecture 9 – Measures of association • 2×2 table (RECAP) • Measures of association – – Risk difference Risk ratio – Odds ratio • Calculation & interpretation of confidence interval for each measure of association
- 9. 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 With outcome (diseased) Without outcome (disease-free) Total Exposed (group 1) d1 h1 n1 Unexposed (group 0) d0 h0 n0 Total d h n
- 10. 2×2 table - Measures of association • Different between measures of association outcome and exposure • Can calculate confidence intervals and test statistics for each measure Measure of Effect Formula Risk difference (lecture 8) p1-p0 Risk ratio (relative risk) p1 / p0 Odds ratio (d1/h1) / (d0/h0)
- 11. 2×2 table – TBM trial example Death during 9 months post start of treatment Treatment group Yes No Total Incidence risk of death (p) Odds of death Dexamethasone (group 1) 87 (d1) 187 (h1) 274 (n1) d1 / n1 = 0.318 d1 / h1 = 0.465 Placebo (group 0) 112 (d0) 159 (h0) 271 (n0) d0 / n0 = 0.413 d0 / h0 = 0.704 Total 199 346 545
- 12. 2×2 table – TBM trial example • Risk difference = p1-p0 = 0.318 – 0.413 = -0.095 (or -9.5%) • Risk ratio = p1/p0 = 0.318 / 0.413 = 0.77 • Odds ratio = (d1/h1) / (d0/h0) = 0.465 / 0.704 = 0.66 Death during 9 months post start of treatment Treatment group Yes No Total Dexamethasone (group 1) 87 (d1) 187 (h1) 274 (n1) Placebo (group 0) 112 (d0) 159 (h0) 271 (n0) Total 199 346 545
- 13. 2×2 table – Calculation of Odds Ratio Commonly given formula for odds ratio (a×d) / (b×c) = (87×159) / (187×112) = 0.66 Death during 9 months post start of treatment Treatment group Yes No Total Dexamethasone (group 1) 87 (a) 187 (b) 274 (n1) Placebo (group 0) 112 (c) 159 (d) 271 (n0) Total 199 346 545
- 14. 2×2 table – Calculation of Odds Ratio Odds ratio for not dying = (a×d) / (b×c) = (187×112) / (=1/0.66) (87×159) = 1.51 Death during 9 months post start of treatment Treatment group No Yes Total Dexamethasone (group 1) 187 (a) 87 (b) 274 (n1) Placebo (group 0) 159 (c) 112 (d) 271 (n0) Total 346 199 545
- 15. Differences in measures of association • When there is no association between exposure and outcome, – risk difference = 0 – risk ratio (RR) = 1 – odds ratio (OR) = 1 • Risk difference can be negative or positive • RR & OR are always positive • For rare outcomes, OR ~ RR • OR is always further from 1 than corresponding RR – If RR > 1 then OR > RR – If RR < 1 the OR < RR
- 16. Interpretation of measures of association • RR & OR < 1, associated with a reduced risk / odds (may be protective) – RR = 0.8 (reduced risk of 20%) • RR & OR > 1, associated with an increased risk / odds – RR = 1.2 (increased risk of 20%) • RR & OR – further the risk is from 1, stronger the association between exposure and outcome (e.g. RR=2 versus RR=3).
- 17. Inference • Obtain a sample estimate, q, of the population parameter (e.g. difference in proportions) • REMEMBER different samples would give different estimates of the population parameter (e.g. sample 1 q1, sample 2 q2,…) • Derive: – Standard error of q (i.e. s.e.(q)) – Confidence interval (i.e. q ± (1.96 × s.e.(q) )
- 18. Ratios – Risk ratio (RR) or Odds ratio (OR) • Usual confidence intervals formula, q ± (1.96×s.e.(q)), is problematic for ratios. When q is close to zero and s.e.(q) large, calculated lower limit of confidence interval may be negative…
- 19. Risk ratio (RR) • Solution Calculate the logarithm of (logeRR) and its standard error RR 1 − 1 1 − 1 s.e.(lo g e RR ) = + d1 n1 d0 n0 95% CI for logarithm of RR :- Upper limit Lower limit = logeRR = logeRR + 1.96×s.e.(logeRR) - 1.96×s.e.(logeRR) 95% CI for Risk ratio (RR):- Upper limit = antilog (upper limit of CI for logeRR) Lower limit = antilog (lower limit of CI for logeRR)
- 20. Log to the base e & antiloge (exponential) • ‘Natural logarithms’ use the mathematical constant, e, as their base, e=2.71828…… 1618 – Scottish Mathematician: John Napierex• antilogex = exp(x) = e = 2.718 loge2.718 = 1 antiloge1 = 2.718 e2 = 7.388 loge7.388 = 2 antiloge2 = 7.388 e3 = 20.079 loge20.079 = 3 antiloge3 = 20.079 101 = 10 log1010 = 1 antilog101 = 10 102 = 100 log10100 = 2 antilog102 = 100 103 = 1000 log101000 = 3 antilog103 = 1000
- 21. 2×2 table – TBM trial example Risk ratio = p1/p0 = 0.318 / 0.413 = logeRR = loge(0.77) = -0.26 0.77 1 − 1 + 1 − 1 s.e.(lo g RR ) =e = 0.11 87 274 112 271 95% CI for logeRR: -0.48 up to -0.04 95% CI for RR: exp(-0.48) up to exp(-0.04) = 0.62 up to 0.96 Death during 9 months post start of treatment Treatment group Yes No Total Dexamethasone (group 1) 87 (d1) 187 (h1) 274 (n1) Placebo (group 0) 112 (d0) 159 (h0) 271 (n0) Total 199 346 545
- 22. Using Stata csi 87 112 187 159 | 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 .9613505 .0386495 .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
- 23. 2×2 table – TBM trial example Interpretation….. Dexamethasone was associated with an estimated decreased risk of 23% (estimated RR=0.77) for death during 9 months post start of treatment. We are 95% confident, that the population risk ratio, lies between 0.62 (decreased risk of 38%) and 0.96 (decreased risk of 4%). Death during 9 months post start of treatment Treatment group Yes No Total Dexamethasone (group 1) 87 (d1) 187 (h1) 274 (n1) Placebo (group 0) 112 (d0) 159 (h0) 271 (n0) Total 199 346 545
- 24. 95% confidence interval for Odds ratio (OR) • Calculate the logarithm of OR (logeOR) and its standard error. 1 1 1 1Woolf’s formula s.e.(lo g OR) =e + + + d1 h1 d0 h0 95% CI for logarithm of OR :- Upper limit = logeOR + 1.96×s.e.(logeOR) Lower limit = logeOR - 1.96×s.e.(logeOR) 95% CI for Odds ratio (OR):- Upper limit = exp (upper limit of CI for logeOR) Lower limit = exp (lower limit of CI for logeOR)
- 25. 2×2 table – TBM trial example Odds Ratio = (d1/h1)/ (d0/h0) = 0.66 logeOR = loge(0.66) = -0.42 1 + 1 + 1 + 1 s.e.(lo g OR ) =e = 0.18 87 187 112 159 95% CI for logeOR: -0.77 up to -0.07 95% CI for OR: exp(-0.77) up to exp(-0.07) = 0.46 up to 0.93 Death during 9 months post start of treatment Treatment group Yes No Total Dexamethasone (group 1) 87 (d1) 187 (h1) 274 (n1) Placebo (group 0) 112 (d0) 159 (h0) 271 (n0) Total 199 346 545
- 26. Using Stata . csi 87 112 187 159, or | 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 Odds ratio | | | | | -.0957659 .7682808 .2317192 .1164974 .6604756 | | | | | -.1762352 .6139856 .0386495 -.0152966 .9613505 .3860144 .4652544 .937623 (Cornfield) +------------------------------------------------- chi2(1) = 5.39 Pr>chi2 = 0.0202 For OR, by default Stata uses Cornfield’s formula for se. You can request the Woolf formula as csi 87 112 187 159, or woolf
- 27. Test statistic for Risk ratio (RR) & Odds ratio (OR) Null hypothesis:- population RR = 1 or population OR = 1 • For risk ratio:- log e RR − log e1 − 0.26 − 0 z = = = −2.4 s.e.(log RR ) 0.11e 2-sided p-value = 0.016
- 28. Test statistic for Risk ratio (RR) & Odds Null hypothesis:- ratio (OR) population RR = 1 or population OR = 1 • For odds ratio:- log e OR − log e1 − 0.42 − 0z = = = −2.3 s.e.(log OR) 0.18e 2-sided p-value = 0.021
- 29. Comparing the outcome measure of two exposure groups (groups 1 & 0) 1 0 1 01 0 s.e.( p ) + s.e.( p ) 1 0 p = Outcome variable – data type Population parameter Estimate of population parameter from sample Standard error 95% Confidence interval for population parameter Numerical µ1−µ0 x1 x0 s.e.( x1 − x 0 ) 2 2 = s.e.( x1 ) + s.e.( x 0 ) x1 − x0 ±1.96× s.e.( x1 − x 0 ) Categorical π1−π0 p − p s.e.( p − p ) 2 2 p − p ±1.96×s.e.( 1 − p0 )
- 30. Comparing the outcome measure of two exposure groups (groups 1 & 0) s.e.(lo g RR ) =e − + − 1 1 0 0 Outcome variable – data type Population parameter Estimate of population parameter from sample Standard error of loge(parameter) 95% Confidence interval of loge(population parameter) Categorical Population risk ratio p1/p0 1 1 1 1 d1 n1 d0 n0 log eRR ±1.96× s.e.(log eRR ) Categorical Population odds ratio (d1/h1) / (d0/h0) 1 1 1 1 s.e.(lo ge OR) = d + h + d + h logeOR ±1.96×s.e.(log eOR)
- 31. Calculation of p-values for comparing two groups z = s.e.( p − p ) z = e s.e.(log (OR )) Outcome variable – data type Population parameter Population parameter under null hypothesis Test statistic Numerical µ1−µ0 µ1−µ0=0 x1 − x0 s.e.( x1 − x 0 ) Categorical π1-π0 Population risk ratio Population odds ratio π1-π0=0 Population risk ratio=1 Population odds ratio=1 z = p1 − p0 1 0 loge (RR) s.e.(log ( RR )) z = loge (OR) e
- 32. Comparing the outcome measure of two exposure groups (TBM trial: dexamethasone versus placebo) Outcome variable – data type Population parameter under null hypothesis Estimate of population parameter from sample 95% confidence interval for population parameter Two-sided p-value Categorical Population risk difference = 0 p1-p0 = -0.095 -0.175, -0.015 0.020 Categorical Population risk ratio = 1 p1/p0 = 0.77 0.62, 0.96 0.016 Categorical Population odds ratio = 1 (d1/h1) / (d0/h0) = 0.66 0.46, 0.93 0.021
- 33. Using Stata – p-value calculated using Chi-squared test . csi 87 112 187 159, or | 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 Odds ratio | | | | | -.0957659 .7682808 .2317192 .1164974 .6604756 | | | | | -.1762352 .6139856 .0386495 -.0152966 .9613505 .3860144 .4652544 .937623 (Cornfield) +------------------------------------------------- chi2(1) = 5.39 Pr>chi2 = 0.0202 For OR, by default Stata uses Cornfield’s formula for se. You can request the Woolf formula as csi 87 112 187 159, or woolf
- 34. Lecture 9 - Objectives • Calculate and interpret the measures of association and their and test statistics confidence intervals – – – Risk difference Risk ratio Odds ratio