521, EP Spring 2007, Vol I, Part 1
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Statistical Methods in Epidemiologic Research
(FOR CLASS USE ONLY DO NOT CITE OR REPRODUCE)
EP 521 Spring 2007 Course Notes Vol I (Part 1 of 5)
A. Russell Localio*, and Jesse A Berlin (The Great Master)
*Department of Biostatistics and Epidemiology Center for Clinical Epidemiology and Biostatistics School of Medicine University of Pennsylvania Philadelphia PA 19104-6021 Statistical Science, Biometrics and Clinical Informatics (BCI). J&J Pharmaceutical Research and Development, LLC 1125 Trenton-Harbourton Road PO Box 200 Titusville, NJ 08560
Overview Some basic principles
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Principle #1 --Perspective This course is about analyzing data -- the issues and methods. It is not about learning rules and applying them to data. Changing Perspectives From Religion to Science After 17 years of interacting with physicians, I have come to realize that many of them are adherents of a religion they call Statistics. ... To the physician who practices this religion, Statistics refers to the seeking out and interpreting of p-values. Like any good religion, it involves vague mysteries capable of contradictory and irrational interpretation. It has a priesthood and a class of mendicant friars. And it provides Salvation: Proper invocation of the religious dogmas of Statistics will result in publication in prestigious journals.
Salsburg DS. The religion of statistics as practiced in medical journals. Am Statistician 1985;39:220-223.
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But this course is also about learning where to find answers rather than just learning what the teacher thinks is best: We should not elevate our autonomy as individual faculty members above every other value. [Professors have a responsibility] to resist the allure of certitude, the temptation to use the podium as an ideological platform, to indoctrinate a captive audience, to play favorites with the likeminded and silence the others. Arenson KW. Columbia chief takes disputes over professors. The New York Times. (March 24, 2005) (Quoting Columbia President Lee C. Bollinger) .
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Principle #2. --Common themes: We relate concepts of epidemiology (and we must be careful about taxonomy) to elements of probability to see some simple methods of analyzing data. We distinguish terms carefully: odds ratio, relative risk, rate ratio, hazard ratio. Each has a mathematical interpretation. They are not generally interchangeable. We try to look at data first and only then implement the statistical tools Interpretation of computer output is not only important, it is essential.
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Principle #3. Applying principles of statistics (from EP 520) to epidemiologic research Principle #4. Common statistical methods include:
a) descriptive data (Means, medians, plots, tables of counts and percentages) Looking at the data b) Statistical test, e.g., t-test comparing cholesterol levels in CHD patients and controls (de-emphasized) c) comparisons of groups, in general. [treated vs. control, exposed vs. unexposed] d) Estimating effect size -- A measure of the effect of treatment or exposure on outcome expressed in any number of different metrics (odds ratios, relative risks, risk differences, differences in means, area under the ROC curve all of which are related.
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Principle #5. Critically important issues - Sampling Variability and power
Sampling Variability and Power Differences between each set of distributions = 12 SD = 4 on left pairs; 2 on right pairs
.2
0
.0 5
y .1
What does the figure suggest? The differences in the two sets of samples are about the same (the difference between the sets of peaks). But peaks in right set have less overlap because they are higher and steeper. Why? Because the variance is smaller.
.1 5 0
12
24
36
48
60
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- Probability distributions (~N(mean, var), or ~B(n,p) ) - Null hypothesis (hypothesis testing) - Confidence interval (meaning and construction) Definition of 95% CI (from frequentist perspective)? (See Woodward, Second Edition p 34) In 1000 repeated samples taken from a population that has a true but unobserved parameter (such as an odds ratio), the estimated 95% confidence interval from each sample should cover the true parameter in 950 of the samples. - Regression ( and the idea of modeling) - All these share underlying goals: Distinguish Signal from noise Systematic differences from random variation
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Principle #6.
Biases in generating data --
Observed sources of bias we can handle by: Exclusion criteria Separate analyses for each group Stratified analyses (Mantel Haenszel methods represent an example) Regression (a statistical model) Unobserved sources: Must undertake sensitivity analysis Ask what would be effect of an omitted confounder that has a stipulated Association with the outcome Association with the exposure of interest How strong must be those associations to make the observed association go away
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Principle #7. Central limit theorem (CLT) is central to our course material for EP 520 and 521 It says that: For largenumbers of subjects, we can use the normal distribution to compare proportions, conduct tests of epidemiologic parameters, and create 95% confidence intervals. The CLT lets us use 2 and Z tests (t tests). (Ref: Woodward p. 72) How large is large depends on : outcome (continuous, binary, counts, modes), number of covariates (exposure + other potential confounders) form of the model (what is treated as fixed) assumptions (more assumptions make model easier to fit but less generalizable)
EP
When we speak (or write) about large sample methods or symptoticmethods, we are referring in general to this concept related to the central limit theorem. As the sample gets large, the statistics we estimate conform to known distributions and we can use the form of those known distributions (normal, chisquare) to make inferences about variance and to estimate confidence intervals. If we cannot invoke large sample theory, then we must use exact methods.
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Log transformations of odds ratios (OR) and relative risk (RR) We transform some measures to enable us to use the central limit theorem to greater effectiveness. Here is an example of why we use log transformations (and why Stata does the same) in estimating confidence intervals. Assume a population of 10,000 patients with outcome y and exposure x. X= Y 0 1 0 1 Total
2,500 2,500 5,000 2,500 2,500 5,000 10,00 0
Total 5,000 5,000 What is the true odds ratio?
The resulting estimate = 1.0 Now, what happens if we draw samples of size 100 from this population and estimate the Ors?
. cs y x, or.
First random sample -- OR = 0.871 For the second
-- OR = 0.714.
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Each sample will have a different OR. What will be the distribution of these OR s? Will we find a symmetric distribution around the true value of 1.0? The distribution is not symmetric about 1.0. We would expect this distribution because an OR = 3 when inverted becomes an OR = 0.33. Each is about equally unlikely to occur upon repeated sampling. To achieve a symmetric distribution, we estimate ln(OR), which should be centered at 0.0 (ln(1.0) = 0.0).
999 Samples of 100
1.5 Density 0 0 .5 1
1
2 Odds Ratio
3
4
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Principle #8. Exact methods vs asymptotic (large sample) methods To avoid having to make assumptions about normal distributions and large samples, there are a set of methods called exact methods or permutation test based or assumption-free methods that do not rely on these assumptions. They work by taking the observed data and calculating a test statistic, such as a chi-square value, but then instead of comparing that test statistic to a chi-square distribution (or other relevant distribution), they take another course. These methods permute (rearrange) the data in all possible ways. With each permutation, the same test statistic is calculated. This exercise results in an entire distribution of test statistics that would occur under all possible rearrangements of the data. One can obtain a p-value by examining where the test statistic from the observed data appears along this distribution. If it appears on one end or the other of the actual distribution, then there are few values of the test statistic that are the same or more extreme than the value for the observed data. This situation equates to a small p-value Confidence intervals can also be obtained, in a very computer intensive manner But exact methods might not always be appropriate or the best estimates . Reference: Good P. Permutation tests. Second Edition. New York: Springer-Verlag 2001
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Principle #9. Frequentist (vs Bayesian) methods These are the subject of this course. They rely on the principle that we could in theory conduct an experiment many times, or we would in an observational setting draw many samples. And each of these samples would differ from the underlying population, and so each estimate differs from the true value, but the goal is to estimate the population parameter whether it be odds ratio (OR), relative risk (RR), hazard ratio (HR), mean, and that true population parameter is assumed to be fixed. The accuracy of a analysis is always evaluated in terms of its performance with repetition. We do not engage in Bayesian inference, although much of our thinking about results takes on Bayesian principles. Bayesians assume that the true population parameter has a distribution. Before the data are collected and analyzed, this distribution is called the prior. After collecting the data, this prior is adjusted to estimate the posterior distribution of the parameter. So, for example, one enters the analysis with a prior estimate of OR=1.0, and then after the data, one adjusts this estimate and its distribution into a posterior based on what the data show. Reference: Moye LA. Statistical Reasoning in Medicine. The Intuitive p-value primer. New York: Springer-Verlag; 2000 (chapter 10)
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1. Basic Concepts of epidemiology and biostatistics (mostly review --Woodward Chap 1) 1.1 Measures of outcome: Know the taxonomy You must say what you mean and mean what you say. This is a fault of many manuscripts. Risk and rate are distinguished by denominators
Risk = Pr [develop disease] for a subject (cumulative incidence) Bounded 0, ... 1. Assumes a closed population Rate = Number developing disease per unit time -- incidence rate (incidence proportion- Rothman, p. 37) Units = person-time Unbounded 0,..., (Woodward pp 117, 152)
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Example of risk (cumulative incidence) and incidence: Incidence of CHD in 12 yrs. (men) e.g., from Framingham ( Schless. p.29) N No. CHD at Risk Events 789 40 742 656 88 130 Cum. Incidence (Risk) .051 .119 .198 Person Yrs Observation 9228 8376 7092 Inc.Rate/ 1000 PY 4.3 10.5 18.3
Age 30 - 39 40 - 49 50 - 59
NOTE: Risk = (e.g.) 40/789 = .051 Rate = [40/9228] x 1000 = 4.3/1000 PY Person years means person years at risk. So, must substract the dropouts from the person years. (Also called the exposure). 9228 789 x 12 = 9468. Each subject who developed CHD not at riskfor entire 12 years. For simplicity, in the example, assumed that those who developed CHD were at risk for 6 years. [In general, we would calculate follow-up
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time using actual individual contributions.] 9228 = (40*6) + (789-40) * 12= 240 +8988 Notes: 1. Assumes constant risk over interval 30 - 39, although risk increases with age. 2. Generally, a 36-year-old man followed for 12 yrs would (should) contribute 4 years to 30 -39 category, and the next 8 to 40 - 49. So, we must distinguish clearly between risk and rate in (1) definition (2) estimation (3) reporting to the reader
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Analysis using "ci" and "cii" commands in STATA for confidence interval: First, incidence rate (per person time). To get estimates and confidence intervals of rates from the Framingham data, we could use Stata as follows
. ci events, exposure (expos) by (agecat)
Where expos is a variable that contains
the person time
-> agecat=30-39 -Poisson Exact -Variable | Exposure events | 9228 -> agecat=40-49 -Poisson Exact -Variable | Exposure events | 8376 -> agecat=50-59 -Poisson Exact -Variable | Exposure events | 7092 Mean .0183305 Std. Err. .0016077 [95% Conf. Interval] .0153151 .0217659 Mean .0105062 Std. Err. .00112 [95% Conf. Interval] .0084263 .0129439 Mean .0043346 Std. Err. .0006854 [95% Conf. Interval] .0030976 .0059025
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Now cumulative incidence (risk): persons with events, per person (Using STATA immediate commands = cii) . cii 789 40 Variable | | . cii 742 88 Variable | | Obs 742 Mean .1185984 Std. Err. .0118693 -- Binomial Exact -[95% Conf. Interval] .0962128 .1440643 Obs 789 Mean .0506971 Std. Err. .0078101 -- Binomial Exact -[95% Conf. Interval] .0364633 .068398
. cii 656 130 Variable | | Obs 656 Mean .1981707 Std. Err. .0155636 -- Binomial Exact -[95% Conf. Interval] .1683193 .2307646
These are basic tools for computing the simple results for the start of your Results section of your manuscript.
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[Aside Simplifying use of Stata For version 9 Many of these commands, such as ci or cii are automated in menus: To get to them, look at the top tool bar find statistics and then from the column of items that come up under statistics look for observational/epi analysis. tables for epidemiologists cohort-study, risk ratio, etc, calculator or case-control Then, fill in the desired table
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Note: Students want to know how to obtain p-values easily once they have a test statistic. A useful add on is quest. This is easily obtained from www.stata.com/quest. It is easily to install within Stata It has statistical tables for computing p-values, for example, for Normal, Chisq distributions Once installed in Version 9, you invoke it by quest on. Then you can use it by going to the toolbar: user/stataquest/calculator/statistical tables] End Aside ]
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Risk and Odds: Comparing groups with different exposures (treatments):
True population D+ D-
E+ A C
EB D
M1 = A + B M0 = C + D
N1 = A + C N0 = B + D Risk = cumulative incidence: R1 = A/N1 R0 = B/N0 Subscript 1 means exposed or diseased 0 means not exposed or not diseased R A/ N1 Relative Risk: RR = 1 = R0 B/ N 0 RR > 1 means exxposed at greater risk RR < 1 means exposed at lower risk (treated are protected)
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A A (A + C) ( A + C) Odds of disease : Among exposed: = R1 /(1- R1 ) = = = A ( A + C A) C 1 (A + C) (A + C) Disease Odds ratio (OR): R /(1- R1 ) A/C AD OR = 1 = = also called the cross product ratio R0 /(1- R0 ) B/D BC OR is important because: 1. For rare diseases, OR and relative risk (RR) and very similar. 2. Exposure OR = Disease OR (via Bayes Rule) in case control studies (We will return to both of these concepts later.) 3. OR is obtained directly from multivariable logistic regression 4. May be estimated from followup or case control study (or even a cross sectional study) 5. IRR (incidence rate ratio). OR is a ratio of incidence rates in density sampling case control design (Rothman, pp. 94-95). Exposure distribution among controls is the same as it is among the persons in the same source population. So, OR = ratio of incidence rates or an IRR (Recall from EP 510)
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We have discussed some simple definitions for event rates: Risk, rate, odds, probability Then we touched briefly on some definitions of effect size Absolute change in risk - risk difference (difference in probabilities), differences in means Relative change in risk relative risk (risk ratio), odds ratio, incidence rate ratio But how large is large when we consider the effect of treatment/exposure?
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EP 521, Spring 2007, Vol I, Part 1 . cs D E [fw=count], or | E | | Exposed Unexposed | Total -----------------+------------------------+-----------Cases | 80 40 | 120 Noncases | 40 80 | 120 -----------------+------------------------+-----------Total | 120 120 | 240 | | Risk | .6666667 .3333333 | .5 | | | Point estimate | [95% Conf. Interval] |------------------------+-----------------------Risk difference | .3333333 | .2140537 .4526129 Risk ratio | 2 | 1.507196 2.653935 Attr. frac. ex. | .5 | .3365161 .6232011 Attr. frac. pop | .3333333 | Odds ratio | 4 | 2.342303 6.830891 (Cornfield) +------------------------------------------------chi2(1) = 26.67 Pr>chi2 = 0.0000 24
Now we relate some of these epidemiologic concepts (and definitions) to probability models.
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But suppose that we are interested in arriving at a measure of effect size that is similar to a nonparametric test of the difference in exposure (E) between two groups characterized by outcome (D)? If the exposures were continuous (e.g, dose of a drug in mg) or least ordered, we might want to do a non parametric test. Recall (Epi 520): Two-sample Mann-Whitney U statistic (equivalent to the Wilcoxon test): Let d ij = 1 if X 1 > X 0 = 0.5 if X 1 = X 0 = 0 if X 1 < X 0 Then for all possible mn pairs of X 1i , X 0 j , where i = 1,..., n; j 1,..., m 1 U= dij , is the fraction of all pairs in which X1 > X 0 and the number of tied pairs mn j i
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This measure of effect is related to others we have seen (or are now seeing). In this case U = 0.667 (1) U = W 0.5m( n + 1) , where W = Rank Sum test statistic nm
Is the effect size large in this case? RD = 0.33, RR = 2.0, OR = 4.0, p< 0.001, all of which might reflect large But U = 0.667, and 0.5 would reflect no difference.
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Another common measure of difference (used heavily in social sciences and psychology): Cohens d: ( x1 x0 ) sd , where sd is the pooled standard deviation
Cohen (and many others) have used the rules of thumb: d size But we can find examples in which this rule makes little sense. 0.2 small 0.5 medium 0.8 large We will return to these issues.
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1.2 Probability Models how they relate to study design What population parameters can we estimate? The population has a parameter, a statistic, such as mean, risk, rate, RD, RR, OR. Efficient estimates for those parameters from samples of data. By efficient we mean that the variances (confidence intervals) are small. What test statistics to use to assess statistical significance (less emphasis in this course).
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[Aside Notation: Persistent problem thoughout the course. Notation is BIG problem with all statistical methods and writings: For your reference. Here are common notations for a 2 by 2 table of cross classified data. E.g. x1 x1+ Both mean summed over second dimension. In this case, it means the sum over the columns to arrive at a row total.
x11 x21 x+1
x12 x22 x+2
x1+ x2+ x++
So, you might see books with this type of notation and you should not be alarmed]
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1. Prospective studies (one margin fixed) Example: Treatment (transfusion) is fixed by design, patients are followed until they develop disease or not. (See also Woodward example 1.2) Transfusion (exposure) Disease Tx+ TxD+ 5 ( x11 ) 13 ( x12 ) D49 35 Column 54 ( x+1 ) 48 ( x+2 ) Sum(fixed) In this example the + denotes summing over the rows, i.e., a column sum. Let PT = Pr [patient in treated group gets infected within 6 months.], and PC = Pr [control infected] Purpose of trial is to test equality of PT and PC. (H0: PT = PC or PT - PC = 0) PT and PC are conditional probabilities.
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PT = Pr[infection|treatment] The probability of infection given that the patient was treated. PC = Pr[infection|control] Usually, we assume X11 and X12 are independent, binomial B(n,p) random variables with parameters (X+1 , PT) and (X+2, PC). Note: X+1 = X11 + X21. Here: P T = X11 / X +1 , and P C = X12 / X +2 RD = Risk Difference = P T - P C . This is the estimate for the parameter of interest. P ) = P (1- P ) / X +2 C Var( C C Var( P T) = P T (1- P T) / X +1
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Can estimate a confidence interval of RD: assuming that (as N get large RD ~ N(RD, Var(RD)) Since we no longer assume H0 is true, we use Var(RD) = Var(P C) + Var(P T ) , assuming independence. p (1- p C ) p T (1- p T ) Var RD = C + X +2 X +1
( )
(Because, VAR(A - B) = VAR(A) + VAR(B) when independence) SE RD = Var RD
( )
( )
95% CI = RD 1.96 SE RD In the example, pT = .093, pC = .271, RD = 0.18 , Z = 2.36 ( p =0.02) 95% CI = (0.03 to 0.33) (Consistent with rejecting H0)
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We can also arrive at a test statistic (to test Ho: RD=0) Under H0: pT = pC = p (common), i.e, RD=0 Where p can be estimated by p= Z=
( X11 + X12 ) and we test H with: 0 ( X +1 + X +2 ) ( pC - pT )
1 1 p 1- p + X +1 X +2
( )
This is pooled variance because we are assuming H0 true. Z ~ N (0,1) is compared to usual standard normal table or to computer function or calculator. This is the test statistic.
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Computing the 95% CI of the risk difference: (no longer assume Ho is true) Distinguish the case of variance under HA. [Aside: In Stata 9. One can fill in the blanks using the statistics option on the tool bar and finding cohort studies . Or one can write out the commands as follows: Recommend using the commands. Fewer errors. Faster to use.]
. csi 5 13 49 45, exact or | Exposed Unexposed | Total -----------------+------------------------+---------Cases | 5 13 | 18 Noncases | 49 45 | 94 -----------------+------------------------+---------Total | 54 58 | 112 | | Risk | .0925926 .2241379 | .1607143 | | | Point estimate | [95% Conf. Interval] |------------------------+---------------------Risk difference | -.1315453 | -.263813 .0007223 Risk ratio | .4131054 | .1577792 1.081613 Prev. frac. ex. | .5868946 | -.0816134 .8422208 Prev. frac. pop | .282967 | Odds ratio | .3532182 | .1214716 1.034842 +----------------------------------------------1-sided Fisher's exact P = 0.0495 2-sided Fisher's exact P = 0.0734
(Cornfield)
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IMPORTANT (THIS CAUSES PROBLEMS FOR MANY STUDENTS) Examples: Odds and risk: Recall: Let risk = p (a probability) Odds = p/(1-p). And p = odds/(1+odds). Essential to know these relationships
Risk of disease | exposed = 5/54. From Stata = 0.092. This is easy and does not confuse. What are the odds of disease | exposure? This causes confusion except at Santa Anita, Gulfstream, Aquaduct, Saratogo and the like. That is odds = [(5/54) / (49/54)] = 5/49 = 0.102. This is greater than the risk. Risk of disease | unexposed = 13/58 = 0.224 from Stata What are the odds of disease | No exposure? (13/45 = 0.29) Much greater than risk.
We have the risk ratio from Stata = 0.41. That = 0.092/ 0.224 a ratio of risks (or probabilities) Odds ratio = ratio of odds = 0.102/0.29 = 0.35 (Also in Stata output) Odds ratio = ad/bc = 5*45/13*49= 0.35. = Cross product ratio.
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So, ad/bc is a common way to get OR quickly, but Be able to compute odds and OR the slow way. Beware of this question on exams = HINT Rare Disease: How Rare is Rare? When is OR a good approximation to RR?
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Pr(D|E+) .002 .01 .02 .06 .08 .10 .12 .16 .18 .20
Pr(D| E-) .001 .005 .01 .03 .04 .05 .06 .08 .09 .10
RR 2 2 2 2 2 2 2 2 2 2
OR 2.002 2.01 2.02 2.06 2.09 2.11 2.14 2.19 2.22 2.25
RR and OR They become different in typical cases. When the baseline or reference group risk is low and RR is modest, RR = OR. But, when the baseline risk increases, and RR remains constant, OR tends to depart from the RR. So, the apparent size of the effect depends on the scale you are using.
This simple concept confuses many investigators.
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Pr(D|E+) .004 .02 .04 .08 .12
Pr(D|E-) .001 .005 .01 .01 .03
RR 4 4 4 4 4
OR 4.01 4.06 4.13 4.26 4.41
What happens when we increase the RR from 2 to 4.
Whether OR is good estimate of RR under the rare disease assumption is dependent on: (1) risk of (D|E-) = underlying disease risk (baseline risk), and (2) RR
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We see that OR and RR are not always the same. A common misrepresentation of results occurs when authors report OR for the situation of (a) high baseline risk (or risk in the reference group (b) large relative risk Example; (one of too many) : Bolton P et al. Comparison of changes in rates of diagnosable major depression before and after intervention JAMA 2003 June 18;289(23):3117-3124. (Prospective study )
Cntl D+ 113 D- 65 178
Pr(D)=159/341 =0.47 Tx 46 159 Pr(D|control) = 113/178 = 0.63 Pr(D|tx) = 46/163 = 0.28 The authors reported OR = 4.4 (controls 117 But the RR = 2.25 So, the OR overstates the 163 341 compared to treated. relative risk two-fold.
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In summary: RR and OR (again) We can compute risk and relative risk and odds and relative odds easily Relative risk is simply the ratio of risks (diseased vs non diseased) Odds ratios is simply the ratio of odds (odds in exposed vs odds in unexposed) Odds = risk only when the risk is very low (close to 0) Odds ratio = risk ratio only when the risk ratio is low AND the risk in the non diseased is low (more about RR and OR later)
Please avoid this pitfall. Remember this difference and the causes for the difference when you are trying to perform power and sample size calculations. The program gives you OR s and you are trying to detect a RR.
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2. Retrospective Studies (such as a case control study) Another margin is fixed: Start with disease status, look backward for exposure For example in a family study: We see the cancer and look back to find exposure. Daughter has Cancer E+ D+ D-
Mother had cancer (EXPOSURE) E7 193 200(X1+) Fixed row 3 197 200(X2+) sums
The probability model here is the same as for the prospective study, i.e., X11 and X21 are independent binomials with parameters (X1+, pCA) and (X2+, pCO). Here: pCA = Pr [exposed|case] pCO = Pr [exposed|control] This what we observe in fact
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What we would like to get (but can=t) would be PE = p (case|exposed) = disease risk PU = p (case|unexposed) But we cannot estimate disease risk directly. Use the following reasoning: Define: RR = pE pU
Assume Rare disease (very important assumption) Then: PE and PU are small, and therefore:
1 pe 1 and 1 pu 1 This assumption means that the odds approximates the risk.
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Then we use the disease OR to approximate the disease RR, we use the fact that the exposure odds ratio = disease odds ratio, and we obtain exposure odds ratio from the data: (1) Disease odds ratio: P E / (1- P E ) P E Pr ( D | E ) = = Obtain approx RR from disease OR (Woodward pp 122-126) P U / (1- P U ) P U Pr D | E
(
)
(2) Obtain Disease OR (DOR )from Exposure OR (EOR) Mother Has Cancer This can be viewed as a table with the typical entries a, b, c and d. E+ ED+ 7 193 D- 3 197 Then, compute the EOR as: (Odds of exposure|disease) / (odds of exposure| no disease) (7/193) / (3/197) = 0.036/0.015 = 2.38
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DOR = (odds of disease|exposure)/ (odds of disease|no exposure) = (7/3) / (193/197) = 2.33/0.98 = 2.38 Pr ( D | E + ) / 1- Pr ( D | E + ) can be re-expressed as: Disease OR = Pr ( D | E ) / 1 - Pr ( D | E ) = Pr( E + | D) / 1- Pr ( E + | D ) = exposure odds ratio,
Pr ( E + | D ) / 1- Pr ( E + | D )
which we can estimate from retrospective data. So, DOR = EOR, odds approximates risk if the disease is rare, and therefore EOR approximates the relative risk.
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3.
Cross sectional study --Table Total Fixed Can estimate all probabilities: Cross sectional study Interest is relationship between infection and tonsils N=1398 British school children had throats swabbed (and cultures) plus tonsils examined.
Neither row nor column sums is fixed by design. Only total number of subjects is fixed. Strep YES Tonsils BIG SMALL 53 19 72
NO 829 497 1326 882 516 1398
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Obtain marginal rates:
Pr (Strep +) = 72/1398 = Pr(S+) Pr (Tonsils big) = 882/1398 = Pr(T+) Pr [S+ and T+] = 53/1398 H0: P(S+ and T+) = Pr(S+) * Pr(T+) (if independent) Note: P(AB)= P(A) * P(B)
Obtain cell probabilities:
Under H0: (in general), expected under independence for the a cell is: 72 882 1398 1398 1398 P(S+ ) P(T + ) N =45.42. X i+ X + j X ++ X ++ X ++ is a test for independence of rows and columns.
E ij =
X i+ X + j X ++
2 Compute usual 1
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(Woodward, pp 47-49). This equation involves the sum of the four cell Eij calculations for the four cells in the table, where j= rows (1 and 2) and j=cols (1 and 2).
i j
12 =
(Oij Eij ) 2
In this sampling framework (and only in this framework)
2 It can be shown that: 1 = Z 2 columns = Z 2 rows
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4.
Both margins fixed - (Table total as well) Seldom true, but may be logical to assume. e.g., small clinical trial N = 20 randomized to treatment or control
Trt Cured Y N 7 3 10 X+1 Ctrl 2 8 10 X+2 9 11 20 X1+ X2+
H0: Assignment to treatment has no effect on outcome. Thus can regard X1+, number cured (under Ho) as fixed ,since the experiment will not affect it, i.e., marginal risk of cure is fixed as we know the overall cure rate
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We know we fixed X+1 and X+2 (treatment assignments) and X++. So what is random? Ans: The number of patients cured who are randomized to treated group, i.e., X11. One cell dictates all the rest. We can compute the remaining cells from the margins. X11 has hypergeometric distribution. X +1 X +2 X11 X12 P ( X11 ) = X ++ X1+ X11 ranges from 0 to min (X1+, X+1) Why? Because margins are fixed.
This distribution forms the basis for Fisher's exact test. (LATER) When X+1 and X+2 become large, p-values for binominal test and Fishers are close.
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We will skip the case when nothing is fixed. (Poisson distribution) Except for an example: Tick Type Deer Location Of trap marsh woods a c Wood b d
Here the length of observation determined total counts and marginal counts. Point: 2 valid regardless of model. We can use the chisq test for Ho: no effect, no association, regardless of the study design.
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1.3 Cohort vs. Case-Control Sampling: We were previously looking at prospective, retrospective, and cross-sectional designs. Another View (flip side of same coin) Cohort: (RCT -- randomized controlled trial): Follow exposed (treated) and non-exposed (control) Pr [D|E] Sampling fractions f1 = fraction of exposed -- e.g., f0 = fraction of unexposed Observed Table (lower case) of cases sampled. Lower case is the number of sampled observations. E+ a c Eb d 1 = 0.1 10 "1 in 10"
D+ D-
m1 m0
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Sampled from population (upper case letters represent population counts) E+ f1A=a f1C=c f1N1=n1 Ef0B=b f0D=d f0N0=n0
D+ D-
m1 m0
Here, we are sampling exposed and unexposed.: f1 = fraction of exposed who are sampled, and f0 = fraction of unexposed who are sampled Pr (D|E)in sample: a A A = f1 = = risk among exposed, subject only to the random variability in the n 1 f 1( A+ C ) N1 sample.
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Pr ( D | E ) =
f 0B = B = risk among unexposed f 0 ( B+ D ) N 0 A/ N1 ad ( f 1 A )( f 0 D ) AD ; OR = = = B/ N 0 bc ( f 0 B )( f 1 C ) BC So, OR is unbiased also.
RR unbiased (clearly) =
So in cohort sampling, we can estimate Risk, RR, OR Recall: OR RR depends on baseline risk. If C N1 D N0 A N1 N 0 AD B CB
Then
So, if disease is rare OR RR
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Case Control Sampling: Recall that the easiest and clearest way to think about a CC study is to regard it as a sample from the entire cohort. Sample cases and controls with sampling fractions g1 and g0; (Before we used f1 for cohort example.) E+ ED+ Da c n0 b d n1 m1 m0
This is the observed sample. These observations come from population (upper case) Now, we are sampling cases and controls, not exposed and unexposed. E+ D+ Dg1A=a goC=c n1 Eg1B=b g0D=d n0 m1 = g1M1 m0 = g0M0
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So cases and controls are sampled at different frequency (and assume sampling does not depend on exposure). Naive Risk Estimate: (E) group a n1 = g1 A A g1 A+ g 0 C N1
This estimate is biased (in the sense that the true value differs from the mean of the estimate over repeated sampling. Systematic rather than random departure of the estimated value from the true value) If g1 = g0, then a n1 = A A = A+ C N1
In practice g1 g0 Both unknown.
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What about estimating OR s from a case control study? E+ D+ a D- c Eb d m1 m0
We can compute unbiased estimate of risk of exposure among cases and controls. P(E | D) = a m1 = g1 A A = g1 A+ g1 B M1 P(E | D)
and exposure odds: and . . .
1- P ( E | D ) ad ( g1 A ) ( g 0 D ) AD OR = = = bc ( g1 B ) ( g 0 C ) BC
So, OR is unbiased estimate for true OR.
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Again, for rare disease OR RR Example: Hill (Proc. R.S. Med, 1965) Death risk in smoking and non-smoking doctors: Risks were: 2270/million among smokers 70/million among non smokers 2270 RR = = 32.4 . True OR = 32.5. Close, because of low event rates 70 Hypothetical full sample: Sampling factors for case-control study E+ D+ D2270 997,730 1 million E70 999,930 1 million Sampling fraction 1/1000 (= g1) (= g0)
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Death risk is low, so 3% (3000 / 100000) = risk, odds The sample then is: E+ ED+ DOR = 1135 998 35 1000
1135 x1000 = 32.5 35 x 998
(Also, see Rothman pp. 95-96, special case in which EOR = incidence rate ratio, without regard to rare disease in case control studies.)
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Example: Using 2 by 2 tables for clinical decision making problem Assume you have a diagnostic test for which you want to estimate the likelihood ratio. You obtain a group of diseased patients and a group of non-diseased. Then you apply a test and obtain T+ or T- (What does this resemble?) Which margin is fixed by design? (The disease status). But we are going to depart from the case/control or cohort or cross sectional design concepts.
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Here are the data (they are binary 0/1 with one row of data per subject) (always remember that the noun data takes a plural verb data are)
. tab T D, col | D T | 0 1 | Total -----------+----------------------+---------0 | 45 8 | 53 | 81.82 18.18 | 53.54 -----------+----------------------+---------1 | 10 36 | 46 | 18.18 81.82 | 46.46 -----------+----------------------+---------Total | 55 44 | 99 | 100.00 100.00 | 100.00
You can compute easily the sensitivity (Sn = Pr(T+|D+) = 0.82), and the specificity (Sp = Pr(T-|D= 0.82). And you could easily obtain a confidence interval about those estimates.
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EP 521, Spring 2007, Vol I, Part 1 . cii 44 36 -- Binomial Exact -Variable | Obs Mean Std. Err. [95% Conf. Interval] -------------+------------------------------------------------------------| 44 .8181818 .0581456 .6728627 .9180797
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We can do this simple calculation of the binomial confidence intervals because the diseased people are different from the nondiseased, and the two samples are independent. The likelihood ratio positive is LR+ = Sn/(1-Sp). LR+ is important because it allows us to compute posterior odds from prior odds. We know from Bayes Theorem (as applied in clinical decisionmaking): Posterior odds = LR * prior odds
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Here LR + = 0.82/(1-.82) = 4.5 Prior probability of disease = 5% (which is a prior odds of 0.05/0.95) =0.053). Then after a positive test, the posterior odds = 0.053*4.5 = 0.24. The posterior probability is then .24/(1+0.24)) = 0.19. What is the confidence interval for 4.5? Set this us as a problem in which we want to compute the relative risk. RR= Sn/(1-Sp) = Pr(T+|D+)/Pr(T+|D-). But this is a RR of obtaining a positive test among diseased and non diseased. (It is backwards from what we might usually think of RR, but the concept is the same).
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Using Stata: we just have to reverse the order of the command to tell Stata to think about this problem as a 2 by 2 table to compute RR from a study in which the test is the outcome and disease is the exposure.
. cs T D | D | | Exposed Unexposed | Total -----------------+------------------------+---------Cases | 36 10 | 46 Noncases | 8 45 | 53 -----------------+--------------...