Cholesterol Reduction Yields Clinical Benefit
A New Look at Old Data
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Abstract
Background There has been a continuing debate about the overall benefit of cholesterol lowering. We performed a novel metaanalysis of all randomized trials of more than 2 years’ duration (n=35 trials) to describe how coronaryheartdisease (CHD), nonCHD, and total mortality are related to cholesterol lowering and to type of intervention.
Methods and Results The analytic approach was designed to separate the effects of cholesterol lowering itself from the other effects of the different types of intervention used. For every 10 percentage points of cholesterol lowering, CHD mortality was reduced by 13% (P<.002) and total mortality by 10% (P<.03). Cholesterol lowering had no effect on nonCHD mortality. Certain types of intervention had specific effects independent of cholesterol lowering. Fibrates (clofibrates, 7 trials; gemfibrozil, 2 trials) increased nonCHD mortality by about 30% (P<.01) and total mortality by about 17% (P<.02). Hormones (estrogen, 2 trials; dextrothyroxin, 2 trials) increased CHD mortality in men by about 27% (P<.04), nonCHD mortality by about 55% (P<.03), and total mortality by about 33% (P<.01). No specific effects independent of cholesterol lowering were found due to diet (n=11) or other interventions (resins, 5; niacin, 3; statins, 2; partial ileal bypass, 1).
Conclusions The results suggest that cholesterol lowering itself is beneficial but that specific adverse effects of fibrates and hormones increase the risk of CHD (hormones only), nonCHD, and total mortality.
Since the mid1960s, almost 50 clinical trials of cholesterol lowering to prevent coronary heart disease (CHD) have been reported. The trials varied in type of intervention (diet, a variety of drugs, surgery), degree of cholesterol lowering achieved, duration of treatment, size, and type of study population. Despite their heterogeneity, the trials have fairly consistently shown a reduction in CHD events (fatal plus nonfatal CHD), and their findings were central to the formulation of guidelines for the treatment of high blood cholesterol.^{1}
However, there is continuing debate about the overall benefit of cholesterol lowering.^{2} ^{3} ^{4} ^{5} ^{6} ^{7} ^{8} ^{9} ^{10} ^{11} ^{12} ^{13} ^{14} Although almost all trials showed a favorable trend for CHD mortality, only 2 reported a significant reduction.^{15} ^{16} Conversely, nonsignificant trends toward increases in nonCHD mortality were observed in 3 trials.^{17} ^{18} ^{19} ^{20} ^{21} Most treatments had no discernible effect on total mortality, exceptions being significant increases in 2 trials^{19} ^{20} ^{21} ^{22} and a significant decrease in 1.^{15} Almost all of the individual trials were designed to examine the effect of cholesterol lowering on CHD incidence and thus lacked the statistical power to examine effects on causespecific or total mortality. In recent years, a number of attempts have been made to answer questions of benefit and risk of cholesterol lowering by the use of metaanalytic techniques that pool the individual trial data.^{23} ^{24} ^{25} ^{26} ^{27} ^{28} ^{29} ^{30} ^{31} ^{32} ^{33} ^{34} Unfortunately, these metaanalyses have not yielded consistent and/or conclusive results for causespecific and total mortality. For example, 4 analyses showed significant reductions in CHD mortality,^{25} ^{27} ^{31} ^{32} ^{33} and 3 did not.^{23} ^{24} ^{26} Five analyses concluded that nonCHD mortality may increase^{23} ^{24} ^{25} ^{26} ^{27} ; 2 did not.^{31} ^{32} ^{34} To date, no metaanalysis has shown any significant effect on total mortality.
Potential reasons for the varying results from metaanalyses include differences in the selection of trials, inadequate power, and heterogeneity among the selected trials. In the present metaanalysis, we attempted to avoid many of these pitfalls insofar as the published data allow. Selection bias was avoided by having very inclusive selection criteria. Statistical power was improved by expressing the results as a function of the degree of cholesterol lowering obtained and by accommodating (and in fact exploiting) the heterogeneity due to type of intervention.
This analysis differs from previous analyses in that it simultaneously addresses two issues of clinical concern: (1) what is the relation of cholesterol lowering to benefit (or harm) and (2) what are the effects of specific types of lipidlowering regimens on clinical outcomes? The clinical outcomes examined are CHD mortality, nonCHD mortality, and total mortality. The central underlying issue is whether the risk reduction (or increase) for each of these outcomes is related to the actual degree of cholesterol reduction or the particular means of reducing cholesterol, not whether it is related to the intention of lowering cholesterol regardless of how that is accomplished. Specifically, the analysis explores whether cholesterol lowering itself (irrespective of the type of intervention used to obtain the cholesterol lowering) may reduce CHD or total mortality, whether cholesterol lowering itself may be harmful, or whether some types of intervention may have negative properties independent of their effect on cholesterol lowering.
Methods
Selection Criteria
The analysis included all the published randomized trials that had the objective of relating cholesterol reduction to changes in mortality or coronary morbidity and that had a duration of ≥2 years. Thirtyfive trials met these criteria. Angiographic trials that met the criteria were included, as were trials that used hormones, surgery, or multifactorial interventions. Trials of short duration (here arbitrarily defined as <2 years) were omitted from the analysis. Trials of short duration contribute little information regarding the effect of cholesterol change on clinical outcomes,^{33} in part because there is a lag time before any effect of cholesterol lowering on CHD can be anticipated.
The relations were evaluated for all 35 trials (unifactorial and multifactorial) listed in Table 1⇓ and for three subsets of these trials: all unifactorial prevention trials (n=31), unifactorial primary prevention trials (n=5), and unifactorial secondary prevention trials (n=26). The outcomes for unifactorial trials are of particular interest because, unlike multifactorial trials, differences between the intervention and control groups can be related to the specific intervention. The primary and secondary intervention trial subsets reflect different target populations for the trials and so provide for the possibility that event rate risk may depend on disease severity on entry into the study.
Clinical Outcomes
Primary analyses were performed for CHD mortality, nonCHD mortality, and total (allcause) mortality. CHD mortality would be expected to reflect the benefit (if any) that can be attributed to the cholesterol lowering or the intervention, whereas nonCHD mortality would be more likely to act as a marker for adverse effects (if any) of cholesterol lowering or the intervention. Total mortality is a robust indicator of overall benefit or risk.
Types of Intervention
We classified the trials by four major types of intervention: diet (11 trials), fibrates (9 trials), hormones (4 trials), and “other” (11 trials). The diet trials fall naturally into one group, and the fibrates were another readily identifiable discrete group of trials that used either clofibrate (7 trials) or gemfibrozil (2 trials). The hormone trials, dextrothyroxine (2 trials) and estrogen (2 trials), were grouped together because they have multiple physiological effects apart from their effects on lipids. The remaining 11 trials, which used resins (5 trials), nicotinic acid (3 trials), lovastatin (2 trials), and surgery (1 trial), were grouped together as “other” trials in the interests of having a manageable number of groups, each of statistically reasonable size.
Statistical Methods
By the likelihoodbased method of trend analysis described in the “Appendix,” risks for each clinical outcome across sets of trials were modeled in terms of degree of cholesterol lowering achieved (slope) and specific effects of type of intervention. For each set of trials (all trials and the three subsets of unifactorial trials), the full model included the slope (same for all the kinds of interventions) and the specific effect of diet, of fibrates, of hormones, and of “other” interventions.
We used a stepdown approach to obtain the best description for each data set analyzed. We chose the most parsimonious model, ie, the one including only the effects found to differ from zero. One result of this strategy was the pooling of intervention categories (usually “diet” and “other”) not found to affect the outcome. Likelihood ratio χ^{2} tests were used to test null hypotheses that the slope and the interventionspecific risks are zero, ie, that there is no trend in event rate with increasing net reduction in serum cholesterol, and that the risks do not depend on the kind of intervention used.^{35} Analyses also were carried out after the hormone studies were omitted, following the same pattern. The result of this process for each analysis was an expression of the logarithm of the odds ratio for the intervention relative to the control for each trial as a sum of the nonzero effects. This expression was then used to predict the outcome of each trial in the model. The values in Tables 2 through 4⇓⇓⇓ are maximumlikelihood estimates for slope and interventionspecific effects across trials.
Statistical significance (“significance” hereafter) refers to twosided tests with α=.05. In addition to predicting the outcomes for each trial, the analyses also address how well the predicted values fit the observed values. A χ^{2} goodnessoffit test is used to assess the adequacy of the resulting models in describing the outcomes observed from the individual trials. A nonsignificant (P>.05) goodnessoffit test result indicates that the model represents the observed data satisfactorily.
Secondary analyses provide, for comparison, the results from conventional pooled odds ratio analyses,^{36} which do not take into account the degree of cholesterol lowering or interventionspecific effects.
The figures illustrate the analytic findings by plotting the observed log odds ratios for event occurrence in each trial on the y axis against the net reduction in cholesterol due to the intervention on the x axis, with the predicted lines relating risk reduction to net cholesterol reduction for each type of intervention where appropriate. The sizes of the symbols on the figures vary inversely with the SD of the log odds ratio and reflect the weight attached to each trial’s outcome due to the numbers of patients at risk and events. Larger studies, with many patients and many events, are weighted more heavily because they estimate the log odds ratio more precisely than smaller studies.
The approximate reduction in risk attributable to an rpercentagepoint net improvement in serum cholesterol is given by the formula
For example, a −0.01 value for slope predicts that a 1percentagepoint net reduction in cholesterol (r=1) would translate into an expected 1% improvement in risk. For treatmentspecific effects,
where ISE is interventionspecific effect. Thus, a value of 0.20 for a specific intervention and a slope of −0.01 would predict that a 20percentagepoint reduction in cholesterol would be needed before an improvement in risk would be observed if that specific intervention were used. An alternative and more intuitive way of expressing this result is that, for a given reduction in cholesterol (here 1 percentage point), the risk is 22% higher due to an interventionspecific effect.
Results
CHD Mortality
Reduced risk of CHD mortality is significantly associated with reduction in serum cholesterol in all trials combined (P<.002), all unifactorial trials (P=.003), and unifactorial secondary prevention trials (P=.009) (Table 2⇑ and Fig 1⇓). In all cases, the slope for cholesterol predicts a 13% to 14% reduction in CHD mortality for every 10percentagepoint net reduction in serum cholesterol. The slope in the unifactorial primary prevention subset was similar, and the lack of significance may be due to the larger variability of the estimate and the small number (5) of primary prevention trials.
The use of hormones is significantly adverse. The estimated hormone effect is about 0.24, suggesting that for any given reduction in serum cholesterol, the CHD mortality risk is 27% higher when the intervention is hormones. The findings predicted from this model fit the observed findings from each trial satisfactorily (goodness of fit, P>.10 in all cases).
The conventional pooled odds ratio analysis does not demonstrate a significant intervention effect for any of the subsets.
NonCHD Mortality
Risk of nonCHD mortality is not significantly related to cholesterol reduction in any of the analyses (Table 3⇑ and Fig 2⇓).
There is a significant adverse effect of fibrate usage for all trials (30% increase, P=.01), all unifactorial trials (29% increase, P=.013), and unifactorial primary prevention trials (39% increase, P=.005). In the secondary prevention trials, the increased risk associated with fibrates is of somewhat lesser magnitude (23%, not significant). The effect of hormone use also is significantly adverse, increasing the risk by about 55% (P<.05). The predicted findings fit the observed outcomes for each trial satisfactorily (goodness of fit, P>.10 in all cases).
The conventional pooled odds ratio estimates indicate a trend toward excess risk in all subsets and are significant for unifactorial trials (19% increase, P<.05) and for unifactorial primary prevention trials (21% increase, P<.05).
Total Mortality
Cholesterol reduction is associated with a lower risk of total mortality in all trials, all unifactorial trials, and in unifactorial secondary prevention trials (P<.05 in all cases) (Table 4⇑ and Fig 3⇓). For every 10percentagepoint reduction in serum cholesterol, the mortality risk is reduced by 8% to 10%. In primary prevention trials, the magnitude of the risk reduction is similar (8%) but is not significant.
Fibrate use is associated with increased total mortality for all trials (17% increase, P=.02), all unifactorial trials (17% increase, P=.03), and unifactorial primary prevention trials (35% increase, P<.02). The 7% increase in unifactorial secondary prevention trials is not statistically significant. Hormones were not used in the primary prevention trials included in our analyses, but in all other sets of trials, their use is associated with increased mortality (32% to 33% increase, P=.01).
A secondary analysis excluding the hormone outcomes provides similar results in regard to the effects of cholesterol reduction and fibrate use. Whether the hormone findings are included or not, the final models all fit the observed data satisfactorily (P>.10 in all cases).
None of the conventional pooled odds ratio analyses detect a significant effect of intervention on total mortality.
Discussion
Effect of Cholesterol Lowering
Taken together, these findings imply that lowering serum cholesterol confers clinical benefit as expressed in a lower CHD mortality and total mortality risk, with the magnitude of the benefit directly related to the degree of cholesterol reduction. This analysis is the first to show that total mortality is reduced by cholesterol lowering. The findings do not lend support to the notion that reducing cholesterol increases the risk of mortality from nonCHD causes, as suggested by some.^{23} ^{24} ^{25} ^{26} If cholesterol lowering were related to nonCHD mortality, one would expect nonCHD mortality to be higher in trials with greater degrees of cholesterol lowering. Instead, the analysis indicates that nonCHD mortality is unrelated to cholesterol lowering, at least for the range of cholesterol values presented by the patients in these trials. A recent extensive evaluation of mortality experience based on cohort studies, international studies, and randomized trials, including unpublished mortality data from the cohort studies, concludes that there is no evidence that low or reduced serum cholesterol increases the risk of nonCHD mortality overall or the risk of death from other circulatory diseases, cancer, accidents and suicide, or other diseases, except for hemorrhagic stroke.^{34}
The data from the Program on Surgical Control of the Hyperlipidemias^{43} contribute materially to the significantly favorable effect of cholesterol lowering on allcause mortality. This is not particularly surprising, in view of the large number of patientyears of followup in this trial and the unusual effectiveness of surgery in lowering serum cholesterol. To evaluate the impact of the POSCH trial, separate analyses of total mortality were carried out (1) excluding the POSCH trial findings and (2) excluding the POSCH trial and the hormone intervention findings. The effect of hormone intervention remained significant at the 5% level after the POSCH findings were excluded, and the effect of cholesterol reduction was nearly significant (P=.07 for all unifactorial trials). The effects of cholesterol reduction and fibrate intervention together on allcause mortality remained significant after both the POSCH and hormone findings were excluded. Our findings for total mortality, therefore, are not due to inclusion of the POSCH study results. Excluding the POSCH trial did not materially affect the favorable relation of cholesterol lowering to CHD mortality.
Effect of Specific Interventions
No effects beyond those associated with cholesterol lowering could be discerned for diet and nonfibrate, nonhormone drugs. These interventions appear to have approximately similar beneficial effects on CHD mortality and total mortality, and their use was not associated with significant adverse effects. However, use of the fibrates employed in the trials included here, clofibrate and gemfibrozil, may entail an enhanced risk of nonCHD mortality that offsets the beneficial effect on CHD mortality and total mortality from cholesterol lowering. The use of hormones in men appears to have a significant adverse effect on all the clinical outcomes studied. This is not unexpected, in view of the findings from the CDP trial,^{22} ^{63} ^{64} even though our analyses do not use the results from the estrogen arms of the CDP trial (because cholesterol reduction for these arms was not reported). The fact that 6 of the 9 fibrate studies (including both gemfibrozil studies) and 3 of the 4 hormone studies, but only 3 of the 17 other studies reporting total mortality, report greater death rates in the treated than in the control groups is consistent with the findings from our analysis. Much of this excess mortality arises from the studies that used clofibrate as the intervention. The findings from the studies that used gemfibrozil are less definitive, because the excess mortality was noncardiac in the primary prevention trial^{18} but cardiac in the secondary prevention trial.^{45}
The nonCHD mortality in the WHO trial is a major contributor to the finding of a significant negative fibrate effect for nonCHD and total mortality. Omitting the WHO trial causes the fibrate effect to lose significance, although the direction remains adverse. However, cholesterol reduction itself remains significantly beneficial for CHD mortality and total mortality. In the United States, clofibrate is now rarely prescribed,^{65} and gemfibrozil is used primarily for lowering triglycerides rather than cholesterol. In the trials reported here, the participants were largely selected for high cholesterol values. Nevertheless, the results remain potentially relevant to practice in the United States because the adverse effects of fibrates do not appear to be related to cholesterol lowering but rather to some other property of the drugs.
If the excess nonCHD and total mortality in the trials were due to the specific effects of the type of intervention rather than to cholesterol lowering, then one would not anticipate a pattern of disease causes across different types of intervention. One would, however, expect to find a pattern within groups of trials that used a specific intervention. There are hints that this may be so for the fibrates and the hormones, although numbers of deaths by specific causes are small. For example, fairly consistent increases in liver and gallbladder disease and in stroke deaths have been found in the larger fibrate studies,^{18} ^{19} ^{20} ^{21} ^{47} ^{48} and the excess of stroke deaths in the 3 trials using clofibrate was found to be statistically significant in a recent metaanalysis.^{66} Excess rates of venous thromboembolism and of CHD have been observed on estrogen interventions,^{63} ^{64} and excess deaths related to cardiac arrhythmias have been observed on dextrothyroxine.^{22} There is a plausible pathophysiological basis for some of these findings (eg, fibrates increase lithogenicity of bile and decrease coagulability of blood, estrogens increase coagulability of blood, and dextrothyroxine promotes cardiac arrhythmias). It has been suggested that hemorrhagic stroke may indeed be related to very low cholesterol,^{34} but it seems unlikely that the levels of cholesterol reached in these trials could cause strokes.
Differences From Previous MetaAnalyses
The findings reported here differ from the results of previous metaanalyses, for two reasons. The trend analysis used here is more directed to the issue of interest (cholesterol lowering) than the conventional pooled odds ratio analysis and, therefore, is more sensitive. Furthermore, we allow for the possibility that different types of interventions may have different inherent risks and that these risk differences may be relevant in assessing mortality experience (as in fact they turn out to be). The enhanced sensitivity of the analysis is best demonstrated by the finding of a significant effect of cholesterol lowering on total mortality, not shown in any of the conventional odds ratio analyses here (Table 3⇑) or in any previous metaanalysis.^{23} ^{24} ^{25} ^{26} ^{27} ^{28} ^{29} ^{30} ^{31} ^{32} ^{33} ^{34} It is also evident in the analyses of CHD mortality (Table 2⇑), in which the result for cholesterol lowering was significant for all but the subset of unifactorial primary prevention trials. The metaanalysis of Law et al^{34} also weighted the odds ratios for degree of cholesterol lowering and was able to show a significant reduction in CHD, but not total, mortality. Possible reasons for the difference from our results are that the analysis by Law et al included trials of <2 years’ duration (which do not offer a good test of the effect of cholesterol lowering) and did not adjust for the heterogeneity introduced by interventionspecific effects.
By simultaneously accommodating the degree of cholesterol lowering achieved and the heterogeneity introduced by different interventions, we were able to improve the fit of the models and to examine whether there were interventionspecific effects. Some previous analyses suggested that drugs (compared with diet or surgery) had more toxic effects, as shown by increased nonCHD and total mortality^{24} ^{25} ^{33} ^{34} and excluded the findings from hormone interventions to remove their specific toxic effects.^{34} We took that suggestion further and were able to identify specific classes of drugs (fibrates and hormones) that account for the adverse outcomes in the drug trials. There was no suggestion that nonfibrate, nonhormone drugs share this risk.
Considering unifactorial primary and secondary prevention trials separately addresses the possibility that the results of intervention might depend on the severity of the patients’ illness on entry into the trial. We found that the direction and magnitude of benefit and risk were similar in the primary and secondary prevention trials, although cholesterol lowering was not significant for CHD and total mortality in the primary prevention trials. However, the fibrate effect was prominent in the primary prevention trials and may explain the excess nonCHD mortality risks found in the analysis by Muldoon et al.^{23}
The metaanalysis of Davey Smith et al^{25} addressed the effect of underlying CHD event rates on response to intervention and concluded that benefit from cholesterol lowering (reduction in CHD mortality) is greatest for patients with high CHD rates, whereas risk (increased nonCHD mortality) is greatest for patients with low CHD rates. These findings are difficult to understand, because there is no biological rationale for cholesterol lowering being beneficial only in the highCHD subset and harmful only in the lowCHD subset. We reanalyzed our data using the classification of risk employed by Davey Smith (more than or fewer than 30 CHD deaths per 1000 patientyears) and found that the benefit of cholesterol lowering in reducing CHD mortality applied to both high and lowCHD subsets. NonCHD mortality was unaffected by cholesterol lowering in either subset but was increased by a hormone effect in the highCHD subset and by a fibrate effect in the lowCHD subset. Our findings suggest that the degree of cholesterol lowering achieved (for CHD mortality) and the type of intervention used (for nonCHD mortality) determine these outcomes, not the underlying risk of CHD.
The findings from our metaanalysis are borne out by the recently reported findings from the Scandinavian Simvastatin Survival Study (4S).^{67} This study is the first largescale secondary prevention trial of a 3hydroxy3methylglutaryl coenzyme A reductase inhibitor using total mortality as the primary end point. Among the 2221/2223 patients assigned to receive simvastatin/placebo, there were 111/189 (5.0%/8.5%) CHD deaths, 46/49 (2.1%/2.2%) nonCHD deaths, and 182/256 (8.2%/11.5%) deaths from all causes. The average serum cholesterol changes from baseline were −25%/+1%, a net benefit of 26 percentage points for the patients on simvastatin. The 42% reduction in CHD mortality observed in 4S is greater than the 29% reduction predicted by our model for a secondary prevention trial with a 26percentagepoint net cholesterol reduction achieved by a nonhormone, nonfibrate intervention. The 30% reduction in total mortality also exceeds our prediction of 20%. NonCHD mortality in 4S was unaffected by cholesterol reduction, as predicted by our model. The 4S findings therefore confirm the predictions of our model that CHD mortality and total mortality will be reduced significantly by effective cholesterol lowering and that nonCHD mortality is not adversely affected by cholesterol lowering achieved with nonhormone, nonfibrate interventions.
Conclusions
The results of the analyses presented here support the concept that cholesterol lowering confers an overall benefit, as shown by the reductions in CHD and total mortality, and that the magnitude of benefit is related to the degree of cholesterol lowering achieved. Patients on hormones appear to be at increased risk for CHD, nonCHD, and total mortality, independent of the degree of cholesterol lowering, whereas patients on certain fibrate interventions, especially clofibrate, appear to have an increased risk of nonCHD and total mortality.
Metaanalyses seldom, if ever, “prove” specific hypotheses. However, the analyses presented here are based on a rational model that describes the data well and, therefore, may generate hypotheses that can be tested by future studies. For example, since excess noncardiac mortality appears to be associated with particular classes of treatment rather than with cholesterol lowering, future studies might focus on possible mechanisms by which these treatments exert their adverse effects rather than on mechanisms by which cholesterol lowering might exert an adverse effect. Our findings also offer some assurance that current guidelines for cholesterol lowering are appropriate, in that they emphasize diet and classes of drugs that are not associated with excess mortality. Cholesterol lowering itself is beneficial and innocuous, provided that effective and welltolerated treatment modalities are chosen.
Appendix A
Trend Analysis
For any trial, let x denote the (average) net improvement in percent cholesterol reduction due to the intervention (intervention minus control), let p_{I} denote the event risk in the intervention group, and let p_{C} denote the event risk in the control group, where
The quantity b is the slope parameter for the trend relating change in risk to net improvement from baseline in serum cholesterol; b=0 if the intervention has no effect. The quantity a can be interpreted as the effect contributed by the study population sampled for the trial. The quantity d expresses the effect of the particular intervention used. The joint effect of the particular intervention and the degree of net cholesterol lowering achieved can be expressed in terms of p_{I} and p_{C},
that is, bx+d is the logarithm of the odds ratio. If p_{I} and p_{C} are fairly small, as in the trials included in Table 1⇑, then the odds ratio can be approximated by the risk ratio, p_{I}/p_{C}. Therefore, 1−e^{bx+d} can be interpreted as the change in event risk due to an xpercentagepoint improvement in cholesterol reduction attributable to the intervention and to the intervention itself. A value <0 means that the risk was reduced; a value >0 means that it was increased.
The likelihood of the outcome in any trial can be expressed as the product of two binomial densities,
where p_{I} and p_{C} are given by Equation 3, n_{I} and n_{C} denote the numbers of patients in the intervention group and control group, respectively, and r_{I} and r_{C} denote the numbers of events among the patients on the intervention and on the control, respectively. This likelihood depends on the parameters a, b, and d. Conditioning on the total number, r=r_{I}+r_{C}, of events occurring among the n=n_{I}+n_{C} patients in the trial removes the dependence on a; the contribution of the kth trial to the likelihood of the sample is the conditional density
where d_{(k)} refers to the intervention used in the kth trial. There are four distinct d_{(k)} values, corresponding to the four types of intervention (fibrates, hormones, diet, and other). Thus, the same d value applies for all fibrate trials, another d value applies for all diet trials, etc. The joint likelihood for a collection of trials is the product of these conditional densities; the value of b remains the same, the value of d_{(k)} depends on the kind of intervention, and the values of the other quantities are trialspecific. The maximum likelihood estimates given in Tables 2 through 4⇑⇑⇑ are the values that maximize the joint likelihood with respect to b and the d parameters. This computation can be accomplished by maximizing the logarithm of the likelihood directly or by using a saddlepoint approximation.^{68} ^{69}
The outcome for each trial can be expressed as a 2×2 frequency table of observed frequencies. The expected frequencies can be computed easily from the predicted log odds ratio for the trial and the marginal totals; there will be four observed and expected frequencies. For the ith of, say, m trials, let S_{i} denote the sum of the four quantities (observed minus expected)^{2}/expected. Also, suppose that the model generating the predicted log odds ratios has k parameters. Then the goodnessoffit statistic for assessing how well the model describes the data is the sum of the S_{i} quantities over the trials, to be compared with a central χ^{2} critical value table with m−k df.
The approach used here for modeling the probability of an event in any trial also arises as a special case of the proportionalhazards model with tied failure times and non–timedependent covariates.^{70} ^{71} In this case, the product of the conditional densities is the partial likelihood for the sample. Finally, the conditional likelihood method (using a multiple noncentral hypergeometric density for the likelihood) can be used in the metaanalysis of epidemiological doseresponse data and provides an alternative to the approaches described by Berlin et al.^{72}
Acknowledgments
We express our gratitude to Carmelo Delgado, PhD, TseWei Chu, and John Cromwell for their efforts in collating and summarizing the study data included in this metaanalysis. We also express our thanks to Dr Jonathan Tobert for several careful readings of the manuscript and many useful suggestions.
Footnotes

Reprint requests to A. Lawrence Gould, Merck Research Laboratories, BL32, West Point, PA 19486.

1 The opinions expressed in this manuscript are those of the authors and do not necessarily reflect the views of the National Institutes of Health.
 Received September 20, 1994.
 Revision received November 28, 1994.
 Accepted December 3, 1994.
 Copyright © 1995 by American Heart Association
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 Cholesterol Reduction Yields Clinical BenefitA. Lawrence Gould, Jacques E. Rossouw, Nancy C. Santanello, Joseph F. Heyse and Curt D. FurbergCirculation. 1995;91:22742282, originally published April 15, 1995https://doi.org/10.1161/01.CIR.91.8.2274
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