Numerous observational studies show that statin use is associated with lower risk of osteoporotic fractures.^{1–5} However, a causal relationship is not supported by data from randomized trials.^{1–5} Healthy-user bias is implicated as a likely culprit for the controversy.^{6–8} It is a type of unmeasured confounding that results from failure to measure and adjust for patient-level tendencies to engage in healthy behaviors. Patients who take statins are more likely to eat a healthy diet, exercise regularly, and avoid alcohol. Failure to account for differences in healthy behaviors between exposure groups yields confounding that overstates the health benefits of statins. One approach reducing healthy-user bias is to adjust for proxy variables that approximate health-seeking behaviors, such as the use of preventative health services.^{6} ^{,} ^{7} If the proxy variables are sufficiently correlated with the confounders, then healthy-user bias will be reduced.^{9}

However, a different explanation for the randomized observational discrepancy is selection bias.^{10} ^{,} ^{11} Most observational studies of statins and fractures included prevalent users of statins within the exposure group rather than restricting the analysis to new (incident) users.^{12} ^{,} ^{13} Prevalent users must survive under treatment for an extended period in order to be included in the study. Because statins reduce the risk of cardiovascular events and mortality, this means that prevalent users will tend to have better health, and hence lower fracture rates, compared with a sample of incident users. This is an example of selection bias because patient health status, and hence fracture risk, is differentially associated with selection among exposed versus unexposed participants.^{10–13} Furthermore, most published studies adjusted for confounders as measured after treatment initiation. The treatment could have influenced the values of the confounders, resulting in further selection bias.^{10}

Both unmeasured confounding and selection bias could play a role in the statins and fracture controversy, but the relative importance of either bias has never been investigated in a quantitative sensitivity analysis. In this investigation, we conduct a systematic review and meta-analysis to summarize the pattern of association between statin use and fracture risk in observational studies. Our objective is to quantify the magnitude of unmeasured confounding and selection bias using Bayesian sensitivity analysis.^{14–17} We conduct a series of analyses using different models for bias. First, we estimate the pooled relative risk for the association between current use of statins and fracture risk in the hypothetical scenario where each study controls for individual-level use of a preventative health service, such as receiving an influenza vaccination. Next, we explore sensitivity of the pooled relative risk as a function of the bias parameters in the prior distribution. Finally, we conduct a sensitivity analysis for selection bias. We conclude with a discussion of the limits of controlling for confounding variables in observational studies of statins and fractures.

METHODS
Review Methods
Our objective was to characterize the pattern of association between current use of statins and fracture risk in all published observational studies. Our review strategy was to update the systematic review of Toh and Hernández-Díaz,^{4} which included 14 observational studies (6 case-control studies and 8 cohort studies) published between 1966 and January 2006. Thus, we assumed that the systematic review of Toh and Hernández-Díaz^{4} was complete.

See Toh and Hernández-Díaz^{4} for discussion of the search strategy and data extraction. We extracted measures of association (relative risk, odds ratio, or hazard ratio) with 95% confidence intervals (CIs) for the relationship between current use of statins and fractures overall. Current use of statins was defined as filling a statin prescription during an exposure time window preceding the fracture date. When more than one adjusted effect estimate was available, the most adjusted estimate was extracted. When an estimate for the association with fractures overall was not available, then we calculated it from the measures of association with each fracture site using Woolf’s method on the logarithm scale.^{18} Studies that focused only on a specific fracture (eg, hip fracture) were included in the overall analysis. For each study, we extracted data on year of publication, study design, sample size, years of follow-up, age and sex of patient population, fracture site, definition of exposure and outcome, inclusion criteria, confounder adjustment, and matching variables.

Statistical Analysis
We calculated the pooled relative risk for the association between current use of statins and fracture risk overall. Additionally, we explored sensitivity to unmeasured confounding and selection bias using the methodology described in McCandless^{14} and in eAppendix 1 https://links.lww.com/EDE/A702 . To illustrate the case of unmeasured confounding, let

or

be an indicator variable for the presence or absence of the confounder at the individual-level in a particular study. For example, U might be an indicator variable for the use of a preventative health service during the statin exposure ascertainment period. Our Bayesian analysis proceeds as follows: each relative risk from included studies has already been adjusted for a set of available covariates (eg, Table 1 ). To additionally adjust for U , we shift the relative risks by adding a fixed amount on the log scale, which is the bias factor

given by

TABLE 1: Effect Measures and Adjustment for Confounding in 17 Observational Studies Included in the Meta-analysis

See McCandless^{14} for discussion of the bias factor

and underlying assumptions. The bias factor

depends on 3 quantities,

,

, and

, which must be chosen by the analyst, and which encode information about the magnitude of confounding from U . The quantity

is the relative risk for association between U and fracture risk, conditional on statin use and measured covariates. The quantities

and

are the prevalences of U among statin users and nonusers, respectively, which we assume do not depend on the measured covariates.^{14}

In contrast, to adjust for selection bias in a meta-analysis, we shift the relative risks from each study by adding the quantity

on the log scale. The quantity

is described in eAppendix 1 https://links.lww.com/EDE/A702 . It is based on the well-known decomposition of the odds ratio in case-control studies with selection bias.^{15–17} The quantity

is the relative probability of selection in fracture cases versus controls, among exposed participants, divided by the corresponding relative probabilities of selection among unexposed participants. It describes the manner in which patient health status, and hence fracture risk, differentially affects participation among prevalent users of statins versus nonusers. See Rothman et al^{15} (equation 19.16) for discussion. Following Greenland,^{19} we assume that the magnitude of selection bias does not depend on measured covariates.

After shifting log relative risk estimates by adding either

or

, we then calculate the pooled relative risk using Bayesian random effects meta-analysis. In practice, both steps proceed simultaneously during Bayesian computation. As described in eAppendix 1 https://links.lww.com/EDE/A702 , we assume that the magnitude of confounding from

and selection bias are the same in every observational study, which means that there is no heterogeneity or secular trends in bias across different studies. For example, we ignore the decline effect, described by Schooler,^{20} which illustrates that effects tend to diminish when studies are repeated over time.

We assign smooth prior distributions to the bias parameters

and

. For

, we assign

where

. These prior distributions are indexed by hyperparameters

, which are chosen by the investigator. The quantities

and

are the prior mean and variance of

on the log scale. The quantities

and

are the prior means and variances of the prevalences

and

on the log odds scale. To explore sensitivity to selection bias, we assign

The hyperparameter

is the prior mean (ie, best guess) for

, whereas the prior variance

describes the uncertainty. Rothman et al^{15} argue that adjustment for selection bias is difficult to address convincingly because there is usually little information available to perform a quantitative analysis. In the absence of information about

, then the quantity

should be large to encompass a broad range of bias.

RESULTS
Search Results and Study Characteristics
Fourteen observational studies published before 2006, and described by Toh and Hernández-Díaz,^{4} were chosen for inclusion.^{2} ^{,} ^{21–30} Our initial search between 2006 and 2012 identified 33 additional articles, of which 28 were excluded based on title and abstract, and a further two excluded upon review of the full text. The three remaining studies^{31–33} were combined with the 14 studies of Toh and Hernández-Díaz^{4} giving a total 17 observational studies of statins and fracture for inclusion in our meta-analysis. The studies are summarized in Tables 1 and 2 . Ten studies controlled for various behavioral risk factors for fractures including smoking,^{2} ^{,} ^{21} ^{,} ^{23} ^{,} ^{25} ^{,} ^{26} ^{,} ^{29} ^{,} ^{32} alcohol consumption,^{21} ^{,} ^{27} ^{,} ^{28} ^{,} ^{32} physical activity,^{2} ^{,} ^{21} ^{,} ^{27} and Vitamin D or calcium supplementation.^{21} ^{,} ^{27} No studies controlled for the use of preventative health services such as recent influenza vaccination.

TABLE 2: Characteristics of 17 Observational Studies Included in a Meta-analysis of Statin Use and Fracture Risk

Pooled Relative Risk Ignoring Unmeasured Confounding and Selection Bias
Figure 1 (solid forest plot) illustrates the relative risks with 95% CIs for the association between current use of statins and fracture risk overall. For two studies^{27} ^{,} ^{33} the relative risk for fractures overall was not available, and so we calculated it from available data. The pooled relative risk (solid diamond) calculated using DerSimonian-Laird was 0.75 (95% CI = 0.66–0.85), and this estimate is nearly identical to that of Toh and Hernández-Díaz.^{4} We calculated

, which is the ratio of heterogeneity to total observed variation, and

on 16 degrees of freedom (P < 0.001).^{34} ^{,} ^{35}

is a test statistic to detect heterogeneity calculated from weighted squared deviations. Tables 1 and 2 reveal many sources of heterogeneity, such as differences in the dose and duration of statin use, differences in study population, and differences in adjustment for confounding.

FIGURE 1: Relative risks for the association between current use of statins and fracture risk. The solid line forest plot indicates 95% CIs extracted from the full text of 17 observational studies chosen for inclusion in the meta-analysis. The dashed line forest plot indicates Bayesian 95% credible intervals, which have been adjusted for influenza vaccination U .

Bayesian Adjustment for Use of Preventative Health Services
We conducted a Bayesian analysis of the forest plot in Figure 1 to adjust the relative risks for the use of preventative health services among individual study participants. Let

be a zero–one variable that indicates whether a study participant obtained an influenza vaccination during the exposure ascertainment period. First, we estimated the prevalence of

among statin users versus nonusers. Brookhart et al^{7} examined statin adherence in relation to the use of preventative health services in n = 20,783 mostly female (86%) patients with average age of 76 years. A total of

received an influenza vaccination over 12 months of follow-up. Patients who filled two or more statin prescriptions during a 1 year ascertainment period were more likely to receive an influenza vaccination (adjusted hazard ratio 1.21, 95% CI = 1.12–1.31). We have

, and we set μ _{p0} = log((x /n )/(1–x /n )) = –0.476

and

as the standard error of the log odds estimator. Following McCandless,^{14} we set

and

The prior distribution is very specific to the study Brookhart et al.^{7} Ideally, it should reflect heterogeneity and uncertainty in the same way that the main random effects meta-analysis allows for heterogeneity.^{34} ^{,} ^{35}

Next, we describe the association between

and fracture risk. An Australian case-control study^{36} of 387 elderly persons (82% female) with mean age 82 years found that engaging in three or more preventative medical practices, such as having a regular medical doctor, dental, eyesight or skin examinations, or an annual influenza vaccination, was associated with lower odds of hip fracture (odds ratio 0.54, 95% CI = 0.32–0.94). Accordingly, we assigned

and

. The receipt of preventative health services is a proxy for behavioral risk factors for osteoporotic fractures. Patients who engage in preventive therapy may for example exercise more, eat a healthier diet, wear a seatbelt when they drive, and avoid tobacco.^{6}

We fit the model of McCandless^{14} using R computer code developed by the author and available in eAppendix 2 https://links.lww.com/EDE/A702 .^{37} The results are given in Table 3 . The right-hand column gives the Bayesian pooled relative risk for the association between current use of statins and fracture risk adjusted for influenza vaccination

. For comparison, the bottom two rows give the pooled relative risk with no adjustment for unmeasured confounding.^{38} Figure 1 (dashed forest plot) shows the shifting of the 95% interval estimates to the right upon adjustment for

. The shifted interval estimate for the pooled relative risk is wider because it incorporates uncertainty from unmeasured confounding.

TABLE 3: Pooled Relative Risk for the Association Between Current Use of Statins and Fracture Risk Adjusted for Different Unmeasured Confounders

Table 3 shows that adjusting for a single binary unmeasured confounder has little impact on the relative risks. In Table 3 , we also adjusted for four other preventative health services using the same methodology to select hyperparameters for the prior distribution. Brookhart et al^{7} reported the association between statin adherence and fecal occult blood test with hazard ratio 1.31 (95% CI = 1.12–1.53), with frequency of the test 2,197 per 20,783 persons. For pneumococcal vaccination, the hazard ratio was 1.46 (95% CI = 1.17–1.83) with frequency of the test 1,184 per 20,783 persons. Dormuth et al^{8} examined the association between statin adherence and screening events in a sample of 141,086 Canadians that was 49% women with average age 61 years. Statin adherence was associated with eye examination with hazard ratio 1.08 (1.05 to 1.12) with frequency of the test 2.93 events per 100 person-years among the less adherent. For sigmoidoscopy, the hazard ratio was 1.07 (0.98 to 1.16) with frequency of test 0.49 events per 100 person-years among the less adherent.

Bayesian Sensitivity Analysis for Unmeasured Confounding
The results in Table 3 depend on the choice of hyperparameters that encode assumptions about unmeasured confounding. We explored sensitivity of the results to changes in the hyperparameters. The magnitude of confounding from

is driven largely by the choice of

and the difference

If

is a large negative number, then a priori we expect that

reduces fracture risk. Similarly, the difference

is the prior mean log odds ratio for the association between

and statin use.

Figure 2 is a contour plot of the Bayesian pooled relative risk after adjusting for

and as a function of

and

The hyperparameter

is fixed inside the interval [−2, 0] to reflect the fact that

reduces the risk of fractures. The difference

is fixed inside the interval [0, 2] so that

is more prevalent in statin users. We fix

and

.

FIGURE 2: Bayesian sensitivity analysis for unmeasured confounding. The contour plot describes the Bayesian pooled relative risk, adjusted for

U , as a function of the hyperparameters

and

in the prior distribution. The solid curve depicts the setting where the Bayesian pooled relative risk is shifted to 1.0 upon adjustment for

U . The dashed curve depicts the setting where the upper bound of the 95% credible interval of the pooled relative risk is shifted to 1.00, which would remove the evidence of a statin–fracture association.

In Figure 2 , the solid curve depicts the scenario in which the Bayesian pooled relative risk is equal to 1.0 upon adjustment for U . For example, one point on the line is

and

, which gives pooled relative risk 1.0. This describes the scenario where the missing confounder

reduces the risk of fractures with relative risk

, and simultaneously,

is associated with statin use with odds ratio

. Thus, large amounts of unmeasured confounding are needed in order to shift the pooled relative risk estimator from 0.73 to 1.0. Less bias is needed to shift the upper bound of the 95% credible interval from 0.73 (0.62 to 0.86) to 0.84 (0.71 to 1.00), which would remove the evidence of a statin fracture risk association.

Calculating the Posterior Distribution for Bias Parameters to Quantify the Healthy-User Bias
The data in Figure 1 can reveal more about the specific nature of the unmeasured confounder. Suppose that statins do not reduce fracture risk, and that the protective association is entirely because of unmeasured confounding. What are the likely characteristics of such a confounder? We can answer this question by repeating the analysis while fixing the prior distribution of the pooled relative risk, adjusted for

, to be a point mass at 1.0, and then studying the posterior distribution that is induced on the bias parameters

. Computationally, this is achieved as follows. First, we set prior distributions for

and

to be point masses at zero, where the pair

are the pooled log relative risk and the between-study variance for the association between statins and fracture risk, respectively. See Section 2.2 of McCandless^{14} for details of notation. Second, we assign

and

to model prior ignorance about the magnitude of confounding from

. Finally, we conduct the Bayesian analysis and study the posterior distribution of the bias parameters

.

Figure 3 illustrates a random sample from the joint posterior distribution of the bias quantities

and

, assuming that the true pooled relative risk for the association between statins and fracture risk is equal to 1.0. The triangle depicts the sample median, which is located at the coordinates (−1.40, 1.38), whereas the sample mean is at the coordinates (−1.66, 1.65). This tells us the following: If the association between statins and fractures is entirely because of unmeasured confounding, then the unmeasured confounder

is likely to have two properties: it reduces fracture risk by roughly 75% because

is estimated to be approximately

, and additionally, it is roughly four times more frequent in statin users because

is approximately

. Other combinations of bias could yield the same results. Because of correlation in the posterior, smaller values of

yield correspondingly larger values of the odds ratio

FIGURE 3: Estimating the magnitude of healthy-user bias from unmeasured confounding. A random sample from the joint posterior distribution of the bias quantities

and

, assuming that the true pooled relative risk for the association between statins and fracture risk is equal to 1.0. The marginal distributions are depicted as a rug plot along the axes. The triangle indicates the sample median.

Bayesian Sensitivity Analysis for Selection Bias
The protective association between statins and fractures may be the result of selection bias because of the inclusion of prevalent users of statins in the analysis.^{10–13} One way to approximately quantify the selection bias is to conduct a meta-analysis separately for the three studies^{26} ^{,} ^{30} ^{,} ^{32} that compared incident users of statins with nonusers (see Table 2 ). Studies that exclude prevalent users are less susceptible to bias. The results are given in the top row of Table 4 . We obtain a pooled relative risk 0.75 (95% credible interval = 0.23 to 2.34). But with only three studies, the predictable effect of pooling a smaller number of studies is simply to widen the interval estimate around the pooled effect estimate.

TABLE 4: Pooled Relative Risk for the Association Between Current Use of Statins and Fracture Risk Adjusted for Selection Bias

To better understand the contribution of selection bias, we conduct a Bayesian sensitivity analysis. First, we specify the hyperparameters

in equation (1) to describe a range of values for the bias parameter

. Among prevalent users of statins, we surmise that participants at lower risk of fractures are more likely to be selected than those at higher risk.^{12} ^{,} ^{13} Conversely, among statin nonusers, there is no relationship between fracture risk and the probability of selection. Thus the bias factor

is greater than 1 and

is greater than 0. We let

vary within the interval

to restrict the range of

that lie between

(no bias) and

. To model uncertainty in

, we set

, which implies a 95% prior credible interval for

that is equal to

. Because three studies^{26} ^{,} ^{30} ^{,} ^{32} excluded prevalent users, these three studies were incorporated into the sensitivity analysis without bias correction.

The results of the sensitivity analysis are given in Figure 4 . Small variations in

suffice to shift the 95% credible interval for the pooled relative risk to cover 1.0 (no association). One strategy to identify reasonable values for

is to use the results of Danaei et al,^{13} who examined selection bias in observational studies of prevalent users of statins among persons with cardiovascular disease. In a meta-analysis of 13 studies that compared prevalent users with nonusers, the pooled hazard ratio for mortality was 0.54 (0.45 to 0.66). In contrast, the corresponding pooled mortality hazard ratio in a meta-analysis of 18 randomized controlled trials comparing statin initiators with noninitiators was 0.84 (0.77 to 0.91). The difference in hazard ratios can be attributed to selection bias, and this gives an indication of the magnitude of bias in the statins and fracture controversy. To adjust for selection bias, we assign a prior for

that approximates the distribution of difference in log hazard ratios. We set

and

, where 0.0977 and 0.0426 are the standard errors of the pooled log hazard ratios. The results are given in Table 4 . The pooled relative risk is shifted close to 1.0, and the association between statins and fractures is eliminated.

FIGURE 4: Bayesian sensitivity analysis for selection bias. The Bayesian pooled relative risk with 95% credible interval as a function of the hyperparameter

. The quantity

_{ψ} is fixed equal to 0.1.

DISCUSSION
According to our analysis, failure to adjust for a single binary unmeasured confounder, such as the use of preventative health services, cannot explain the protective association between statins and fracture risk that has been observed in the literature. In 17 published studies, the pooled relative risk was 0.75 (0.66 to 0.85). Upon adjustment for five different unmeasured confounders, the pooled relative risk shifted by less than 5% on the log scale. This finding is consistent with the conclusions of Cummings and Bauer^{5} and Pasco et al,^{21} who argued that the association between statins and fracture risk was unchanged after adjusting for confounding using various models. If indeed statins do not reduce fracture risk and the associations in Figure 1 are entirely because of unmeasured confounding, then the magnitude of confounding is very large. We estimate that a binary unmeasured

would need to reduce fracture risk by 75% and be four times more frequent among statin users. This suggests that measurement and adjustment for additional fracture risk factors beyond those reported in Table 1 would not materially change the protective trend observed across the 17 studies.

A different explanation for the association between statins and fractures is selection bias because of the inclusion of prevalent users in the analysis.^{10–13} Selection bias occurs when study participation is affected by the exposure or causes of the exposure, and additionally, by the outcome or causes of the outcome. Prevalent users of statins must survive under treatment for an extended period in order to be included in the study. But statins prevent heart disease and cardiovascular events. Consequently, prevalent users will be healthier and have lower fracture risk compared with a sample of incident (new) users, and this may induce selection bias. Table 4 and Figure 4 summarize the results of a sensitivity analysis for selection bias. The results show that moderate amounts of bias, typical of those observed Danaei et al,^{13} could eliminate the association between statins and fracture risk.

There are also other possible sources of bias. There could be several unmeasured behavioral risk factors that are acting in unison (in the sense that they are correlated with one another) and additionally with the exposure and outcome. However, speculating quantitatively about whether this is a reasonable hypothesis is challenging because it is difficult to estimate the size of the correlation between unmeasured confounders.^{9} ^{,} ^{39} Publication bias may also play a role. Toh and Hernández-Díaz^{4} concluded that the preferential publication of studies with positive associations may contribute to the apparent protective effect.

In conclusion, our investigation reveals that regression adjustment for confounding is unlikely to reduce healthy-user bias in observational studies of statins and fractures. This finding is important because much of the debate in the literature centers on the role of various covariates that may or may not have been measured in different studies.^{1} ^{,} ^{3} ^{,} ^{5} Instead, it appears that reducing healthy-user bias requires a focus on design strategies to limit selection bias, such as the new user design that excludes prevalent users from the analysis.^{12}

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