(Circulation. 1995;91:2274-2282.)
© 1995 American Heart Association, Inc.
Articles |
Cholesterol Reduction Yields Clinical Benefit
A New Look at Old Data
From Merck Research Laboratories, West Point, Pa (A.L.G., N.C.S.,
J.F.H.); the National Institutes of Health, Bethesda, Md
1
(J.E.R.); and Bowman Gray School of Medicine, Winston-Salem, NC
(C.D.F.).
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Abstract |
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 Top Abstract IntroductionMethods Results Discussion References |
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Background There has been a continuing debate about the overall benefit of cholesterol lowering. We performed a novel meta-analysis of all randomized trials of more than 2 years' duration (n=35 trials) to describe how coronary-heart-disease (CHD), non-CHD, 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 non-CHD mortality. Certain types of intervention had specific effects independent of cholesterol lowering. Fibrates (clofibrates, 7 trials; gemfibrozil, 2 trials) increased non-CHD 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), non-CHD 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), non-CHD, and total mortality.
Key Words: cholesterol • meta-analysis • mortality
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Introduction |
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TopAbstractIntroductionMethods Results Discussion References |
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Since the mid-1960s, 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 non-CHD 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 trials19 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 cause-specific 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 meta-analytic techniques that pool the individual trial data.23 24 25 26 27 28 29 30 31 32 33 34 Unfortunately, these meta-analyses have not yielded consistent and/or conclusive results for cause-specific 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 non-CHD mortality may increase23 24 25 26 27 ; 2 did not.31 32 34 To date, no meta-analysis has shown any significant effect on total mortality.
Potential reasons for the varying results from meta-analyses include differences in the selection of trials, inadequate power, and heterogeneity among the selected trials. In the present meta-analysis, 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 lipid-lowering regimens on clinical outcomes? The clinical outcomes examined are CHD mortality, non-CHD 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.
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Methods |
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TopAbstractIntroductionMethods Results Discussion References |
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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.
Thirty-five 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.
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Clinical Outcomes
Primary analyses were performed for CHD
mortality, non-CHD
mortality, and total (all-cause) mortality. CHD mortality would be
expected to reflect the benefit (if any) that can be attributed to the
cholesterol lowering or the intervention, whereas non-CHD 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 likelihood-based 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 step-down 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 
2 tests were used to test null
hypotheses that the slope and the intervention-specific 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
maximum-likelihood estimates for slope
and intervention-specific effects across trials.
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Statistical
significance ("significance" hereafter) refers to
two-sided tests with 
=.05. In addition to predicting the outcomes
for each trial, the analyses also address how well the predicted values
fit the observed values. A 2 goodness-of-fit test
is used to assess the adequacy of the resulting models in describing
the outcomes observed from the individual trials. A nonsignificant
(P>.05) goodness-of-fit 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 intervention-specific 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 r-percentage-point net improvement in serum cholesterol is given by the formula
 | (1) |
For example, a -0.01 value for slope predicts that a 1-percentage-point net reduction in cholesterol (r=1) would translate into an expected 1% improvement in risk. For treatment-specific effects,
 | (2) |
where ISE is intervention-specific effect. Thus, a value of 0.20 for a specific intervention and a slope of -0.01 would predict that a 20-percentage-point 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 intervention-specific effect.
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Results |
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TopAbstractIntroductionMethods Results Discussion References |
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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 10-percentage-point 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.
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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.
Non-CHD Mortality
Risk of non-CHD mortality is not
significantly related to
cholesterol reduction in any of the analyses (Table 3 and Fig
2).
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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 10-percentage-point
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.
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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.
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Discussion |
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TopAbstractIntroductionMethods Results Discussion References |
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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 non-CHD causes, as suggested by some.23 24 25 26 If cholesterol lowering were related to non-CHD mortality, one would expect non-CHD mortality to be higher in trials with greater degrees of cholesterol lowering. Instead, the analysis indicates that non-CHD 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 non-CHD 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 Hyperlipidemias43 contribute materially to the significantly favorable effect of cholesterol lowering on all-cause mortality. This is not particularly surprising, in view of the large number of patient-years of follow-up 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 all-cause 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 non-CHD 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 trial18 but
cardiac in the secondary prevention trial.45
The non-CHD mortality in the WHO trial is a major contributor to the finding of a significant negative fibrate effect for non-CHD 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 non-CHD 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 meta-analysis.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 Meta-Analyses
The findings reported
here differ from the results of previous
meta-analyses, 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
meta-analysis.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 meta-analysis of Law et al34
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
intervention-specific 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 intervention-specific effects. Some previous analyses suggested that drugs (compared with diet or surgery) had more toxic effects, as shown by increased non-CHD and total mortality24 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 non-CHD mortality risks found in the analysis by Muldoon et al.23
The meta-analysis of Davey Smith et al25 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 non-CHD 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 high-CHD subset and harmful only in the low-CHD subset. We reanalyzed our data using the classification of risk employed by Davey Smith (more than or fewer than 30 CHD deaths per 1000 patient-years) and found that the benefit of cholesterol lowering in reducing CHD mortality applied to both high- and low-CHD subsets. Non-CHD mortality was unaffected by cholesterol lowering in either subset but was increased by a hormone effect in the high-CHD subset and by a fibrate effect in the low-CHD subset. Our findings suggest that the degree of cholesterol lowering achieved (for CHD mortality) and the type of intervention used (for non-CHD mortality) determine these outcomes, not the underlying risk of CHD.
The findings from our meta-analysis are borne out by the recently reported findings from the Scandinavian Simvastatin Survival Study (4S).67 This study is the first large-scale secondary prevention trial of a 3-hydroxy-3-methylglutaryl 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%) non-CHD 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 26-percentage-point net cholesterol reduction achieved by a nonhormone, nonfibrate intervention. The 30% reduction in total mortality also exceeds our prediction of 20%. Non-CHD 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 non-CHD 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, non-CHD,
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 non-CHD and total
mortality.
Meta-analyses 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 well-tolerated treatment modalities are chosen.
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Acknowledgments |
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We express our gratitude to Carmelo Delgado, PhD, Tse-Wei Chu, and John Cromwell for their efforts in collating and summarizing the study data included in this meta-analysis. We also express our thanks to Dr Jonathan Tobert for several careful readings of the manuscript and many useful suggestions.
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Footnotes |
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Reprint requests to A. Lawrence Gould, Merck Research Laboratories, BL3-2, 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. 
Trend Analysis
For any trial, let x denote the
(average) net improvement in
percent cholesterol reduction due to the intervention (intervention
minus control), let pI denote the event risk in the
intervention group, and let pC denote the event risk in the
control group, where
 | (3) |
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 pI and pC,
 | (4) |
that
is, bx+d is the logarithm of the odds ratio. If
pI and pC are fairly small, as in the trials
included in Table 1, then the odds ratio can be approximated by
the
risk ratio, pI/pC. Therefore,
1-ebx+d can be interpreted as the change in
event risk due to an x-percentage-point 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,
 | (5) |
where
pI and pC are given by Equation 3,
nI and nC denote the numbers of patients in the
intervention group and control group, respectively, and rI
and rC 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=rI+rC, of events occurring
among the n=nI+nC 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
 | (6) |
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 trial-specific. 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 2x2 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 Si 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
goodness-of-fit statistic for assessing how well the model describes
the data is the sum of the Si quantities over the trials,
to be compared with a central 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 proportional-hazards model with tied failure times and non–time-dependent 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 meta-analysis of epidemiological dose-response data and provides an alternative to the approaches described by Berlin et al.72
Received September 20, 1994; revision received November 28, 1994; accepted December 3, 1994.
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