Meta‐analytic approaches to determine gender differences in the age‐incidence characteristics of schizophrenia and related psychoses

Abstract A recent systematic review and meta‐analysis of the incidence and prevalence of schizophrenia and other psychoses in England investigated the variation in the rates of psychotic disorders. However, some of the questions of interest, and the data collected to answer these, could not be adequately addressed using established meta‐analysis techniques. We developed a novel statistical method, which makes combined use of fractional polynomials and meta‐regression. This was used to quantify the evidence of gender differences and a secondary peak onset in women, where the outcome of interest is the incidence of schizophrenia. Statistically significant and epidemiologically important effects were obtained using our methods. Our analysis is based on data from four studies that provide 50 incidence rates, stratified by age and gender. We describe several variations of our method, in particular those that might be used where more data is available, and provide guidance for assessing the model fit. Copyright © 2013 John Wiley & Sons, Ltd.


Introduction
A recent systematic review and meta-analysis of the incidence and prevalence of schizophrenia and other psychoses in England (Kirkbride et al., 2011(Kirkbride et al., , 2012) used a range of meta-analytic techniques, including random effects modelling, meta-regression and multivariate metaanalysis, to inspect variation in the incidence and prevalence of psychotic disorders. However, some of the questions of interest, and the data collected to answer these, could not be adequately addressed using established meta-analysis techniques. The aim of this paper is to present the methodological approach that we developed to overcome these difficulties, and provide details of the results. Although the new methodology was applied to several different psychotic syndromes, here we will describe and present results for only one outcome measure, the incidence of schizophrenia.
Gender differences in the incidence and prevalence of schizophrenia are well established from individual surveys (Angermeyer and Kuhn, 1988;Flor-Henry, 1985;Kirkbride et al., 2006). Typically, incidence peaks in the early twenties for men and a few years later for women. Until the mid-thirties, rates are typically estimated to be approximately 1.5 to two times greater in men than women. After this time, rates decline for both sexes, with a narrowing sex ratio, until the mid-forties when there is a smaller secondary peak for women. Such variation may have important implications for understanding the underlying aetiology of the disorder (Hafner et al., 1993), particularly with regard to a putative neuroprotective role for estrogen prior to menopause or the biopsychosocial impact of such experiences. We wished to investigate these differences using the combined precision and power available from a systematic review. In particular, we wished to investigate whether there was evidence of an elevated incidence rate in men compared to women, and later in life for women compared with younger women.

Methodology
Four suitable data sources were identified from the systematic review (Kirkbride et al., 2012). Three of these sources came from published studies (Kirkbride et al., 2006;Brewin et al., 1997;Goldacre et al., 1994), with the final dataset coming from data unpublished at the time of the report (since published; Reay et al., 2010) These sources provided data which motivated the development of our approach to studying sex and age-related variations in incidence in a more flexible manner. Here, each primary study provided the natural logarithm of the raw (unadjusted) incidence rate and its standard error (s.e.) for the study population, for men and women separately, stratified by age group (typically deciles of age). The logarithm of the incidence rate is a standard measure used in meta-analysis for which a normal approximation is conventionally used (Sutton et al., 2000). Eligible studies, however, did not all use the same age strata when reporting incidence rates so age mid-points were used when modelling the data (see Tables 1 and 2).
The data required for analysis are shown in Table 2. From such data we wanted to perform a single metaanalysis to obtain simultaneous statistical inferences to address the following questions: (1) Is there evidence of a sex difference in rates?
(2) Is there evidence of changed rates for women during mid-life? (3) Are rates for men and women more similar in later life than at younger ages?
Our statistical methods were variations and novel applications of new methodology introduced in a recent paper by Rota et al. (2010). Our trends were over the population's age, whereas those of Rota et al. (2010) were over the drug's exposure level, but the statistical issues are similar because, in both cases, flexible trend modelling in the context of meta-analysis was desired. In our application, given prior knowledge in this field, a linear model for the effect of age was inappropriate. Furthermore a simplistic and incorrect assumption e.g. that of linearity, here might adversely influence our inferences for the effects described in the introduction that are of real interest. Instead, we wanted a flexible model of the effects of age, using as few parameters as possible, because our analysis was based on only 50 rates (Table 2). Because the age distribution of schizophrenia rates is often reported to exhibit a secondary peak onset (SPO) in women during their mid-forties, we sought to examine evidence for such variation around this period by explicitly modelling this in our meta-regression. Here we defined 45 years to be the mid-point for the SPO.
We decided, a priori, to avoid a two stage approach such as that of Rota et al. (2010) because this involves estimating the between-study variance structure using a very small number of studies. For our data, with a relatively small number of contributing studies, it is clear that between-study variance parameters estimates will be very imprecise. Two-stage approaches are sometimes advocated in the context of meta-analysis with large datasets (The Fibrinogen Studies Collaboration, 2009), but with just 50 rates to model this is not tenable. Estimation in the second stage of Rota et al.'s (2010) approach depends directly on this estimated variance structure and problems with using random effects models in meta-analyses, even in the univariate case, when there is a small number of studies is now generally appreciated (e.g. Jackson and Bowden, 2009). We further desired a simple and computationally straightforward approach because we intended to use the methodology routinely for a variety of outcome measures (i.e. different psychotic disorders) relevant to the main systematic review (Kirkbride et al., 2012).
Hence we used the model: where Y it was the tth logarithm of the crude (i.e. unadjusted) incidence rate from the ith study and x it was the midpoint of the corresponding age stratum. The terms b 1 x it p 1 þ b 2 x it p 2 were the fractional polynomial used to model age trends in the data (see later). The variables m it and n it were indicators for the corresponding incidence rate for men, and women with an age-stratum midpoint of at least 45 years, respectively. Hence d denoted one parameter of primary interest, a gender effect, and f denoted the other, an effect (raised rate) in women after Meta-analytic approaches to determine gender differences Jackson et al.
the SPO. In situations where the SPO is thought to occur at alternative ages in different studies, n it would be defined as an indicator for women with an age-stratum midpoint that exceeds the study specific SPO. Although we use the term "effect" when describing our parameters, it is important to be clear that our data are observational and no causal associations are implied by this term. Positive d would indicate that women had lower incidence rates than men until the SPO. Positive f would indicate that there was an elevated risk of schizophrenia in women after the SPO compared with women immediately before this. However, a positive f would not imply that older women had higher incidence rates than younger women; it is known that the peak age of onset for schizophrenia is most common in early adulthood. This is because both study and age were included in Equation 1, so estimates of d and f were adjusted for these. In particular, the fractional polynomial in Equation 1 modelled the effect of age. This can, for example, model incidence rates that are strongly decreasing for women as their age increases, so that older women have lower rates than younger women, despite a positive f and hence a step change increase in the incidence rate around the SPO. This can be assessed by examining the model's predictions, as shown in Figures 1 and 2. When drawing inferences we focused on the estimates of d and f , because they address the questions that particularly interested us. However, depending on the questions of interest, any aspect of the model fit could be used for interpretation, including the a i parameters. In addition to enabling us to make the inferences we require, Equation 1 provides a way to explain why the observed rates differ.
The a i were fixed effects, stratified by study, and model the inevitable association between incidence rates from the same study (e.g. Salanti et al., 2008). Since the pooled or study specific rates were of little interest, with d and f being primary, we chose to model the study effects in the simplest and most convenient way for our purposes. Although the a i allowed for between-study heterogeneity, in the sense that different studies were permitted to have elevated or reduced rates, the model assumes common fractional polynomials, gender effects d and SPO effects f in women across studies. If these assumptions are not true then the resulting standard errors will, in general, be too small. We return to this issue in the discussion. However our fixed effect approach has the advantage that the model is easier to interpret because we do not interpret regression coefficients as average values, as we do in random effects modelling.
The term e it was the normally distributed error, with variance taken from study reports. These variances were regarded as fixed and known, a priori, as is conventional in meta-analysis (e.g. Biggerstaff and Tweedie, 1997;Hardy and Thompson, 1996). Their square-root, the standard errors (s.e. values), are shown in Table 2.

Fractional polynomials
Fractional polynomials (Royston and Altman, 1994) are an established way to parsimoniously model trends. We adopt the convention, also used by Rota et al. (2010), of selecting p 1 and p 2 from a predefined set P = {À2, À1 ,À0.5, 0, 0.5,1, 2, 3}, a set of 36 trend functions, including U shaped and J-shaped relations. Powers of p 1 = 0 and p 2 = 0 were taken to indicate taking the natural logarithm. If p 1 = p 2 then x it p 2 is replaced by x it p 2 log x it ð Þ in the fractional polynomial (Royston andAltman, 1994, Rota et al., 2010), so that the regression parameters are identifiable in such instances. We follow Rota et al.   Table 2, are also shown, where the upper and lower bounds of the intervals are connected using dashed lines to aid interpretation. Meta-analytic approaches to determine gender differences Jackson et al.
(2010), in choosing p 1 and p 2 as the best fitting fractional polynomial (minimum weighted sum of squared residuals, where the weights are those used when fitting Equation 1) over all studies and basing all inference on the resulting model, ignoring the uncertainty in the form of the fractional polynomial. This followed the same principle as Rota et al. (2010), who determined the best fitting fractional polynomial using the Akaike information criterion (AIC).

Meta-regression
Meta-regression (Thompson and Sharp, 1999) is a well established technique. Once the form of the fractional polynomial has been chosen, Equation 1 has the same form as a standard fixed effects meta-regression. Furthermore it is a simple extension of Equation 1 of Thompson and Sharp (1999), which included just a single covariate. Hence, once p 1 and p 2 have been determined, Equation 1 can be fitted using a standard linear regression, where the weights are given by the reciprocal of the variances of e it . The correct standard errors of the regression coefficients were, however, obtained by dividing those given by the square root of the reported mean square error (as explained by Thompson and Sharp, 1999). Our Equation 1 was therefore very easy to fit in any statistical software for standard linear models.

Implementation
R software was used to implement the method. First, each possible combination of powers p 1 and p 2 was used in the fractional polynomial when fitting Equation 1. The model that minimized the resulting weighted sum of squares was taken as the best fitting model. Equation 1 was fitted as a standard (weighted) linear regression model using the lm command, with the weights defined as the reciprocal of the variances and where the response variable was the logarithm of the incidence rate. The best fitting model was then used for inference, where the standard errors were corrected as explained earlier.

Results
The best fitting fractional polynomial for the incidence of schizophrenia data was p 1 = -1 and p 2 = -1. This provided a weighted sum of squared residuals of 63.39. Fitting Equation 1 with p 1 = p 2 = -1 required estimating eight parameters (four a i terms, b 1 , b 2 , d and f) from 50 rates, emphasizing the case for our parsimonious model. The resulting estimates wered ¼ 0:69 (s.e.: 0.08) andf ¼ 0:71 (s.e.: 0.18). Confidence intervals can be obtained as parameter estimates plus and minus 1.96 standard errors and similarly p-values can be calculated from these results.
Both estimates resulted in the inferences we anticipated with the interpretation of these coefficients as follows: the estimate of d was positive and highly statistically significant, indicating that the incidence of schizophrenia was greater for men than women prior to the SPO; the corresponding point estimate of the incidence ratio was approximately two. The estimate of f is of similar magnitude and was also statistically significant, but this parameter is less precisely estimated. This relative imprecision seems reasonable, because only nine of the 50 rates are taken from women after the SPO, so this effect must be harder to identify. Nevertheless, this analysis suggests that the incidence ratio for women just before, and just after, the SPO is also around two.
The difference between the log incidence rate of schizophrenia in men and women after the SPO is d À f.
The estimate of this difference wasd Àf ¼ À0:02, and the standard error of this estimated difference was ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Vard þ Varf À 2Covd;f r . The correlation of the estimates of d and f was obtained when fitting Equation 1 as a weighted linear regression as explained earlier, and so their covariance and hence the standard error of the estimated difference was calculated as 0.17. The data did not provide enough information to draw any firm conclusions about the direction of the difference between incidence rates of schizophrenia for men and women in the oldest age groups, but it seems reasonable to infer that these are more similar than those for younger people.
A considerable advantage of our methods was that, since they were based on standard linear models, the examination of model diagnostics such as leverage, influence and residuals are straightforward. Such diagnostics are also considered important in meta-analysis (Viechtbauera and Cheung, 2010). We calculated (approximately) standardized residuals as the ratio of the raw residuals and the studies' standard errors in Table 2. If Equation 1 fits well then these standardized residuals should resemble a standard normal distribution. A histogram of the 50 standardized residuals is shown in Figure 1. The residuals were slightly skewed, but Figure 1 reassures us that the model fit was quite reasonable for these data.
Plots of the observed, and fitted, rates were shown separately for men and women in each of the four studies in Figures 2 and 3, respectively. A comparison of these Jackson et al.
Meta-analytic approaches to determine gender differences plots supported the formal statistical inference that rates were higher in young men than young women. Further, this difference appeared to be reduced in the older population. Although the model captured the general trends in the data quite well, it did not capture the apparent fall in the incidence rate for men in their forties, and subsequent increase in their fifties, that three of the four studies were suggestive of ( Figure 2). If this is a real phenomenon and we wish to describe this, further modelling is required. It is perhaps unfortunate that these unexpected falls occurred at around the age were the model provides a step change for women. The apparent discrepancies between the observed and fitted rates for men in their forties are probably due to a combination of the problems associated with modelling real data and our relatively simple model. The overall impression of Figures 1-3 is that our modelling has performed well, but further refinement may be possible.

Discussion
We have developed a novel methodology in order to flexibly model variation in incidence rates for schizophrenia.
Specifically, this involved combining fractional polynomials with meta-regression. Importantly our method avoided the difficulties associated with attempting to estimate any between-study variance parameters. However, it achieved this by assuming common fractional polynomials, gender, and SPO in women, effects across studies. These are quite strong assumptions but the resulting model adequately describes all the datasets in our review.
The simplest way to relax these assumptions is to adopt a two stage approach and fit Equation 1 to each study in turn and then combine the resulting estimates in a conventional random effects meta-analysis at a second stage. With only four studies, the resulting estimation of the betweenstudy variation here is very imprecise but for examples with more studies this is a more viable solution to allow heterogeneous effects of interest. Despite our reservations in doing so, we fitted univariate random effects models to the four study specific estimates of d and f in order to explore the implications of this alternative procedure. The resulting random effects model for f collapsed to a fixed effects model and similar inferences were obtained, but the random effects model for d provided a large I 2 statistic (Higgins and Thompson, 2002) of 57%, despite the lack of a strongly statistically significant hypothesis test for Figure 3. Female observed (solid points) and fitted (hollow points) log incidence rates in each of the four studies. Ninety-five per cent confidence regions, obtained from the rates and standard errors shown in Table 2, are also shown, where the upper and lower bounds of the intervals are connected using dashed lines to aid interpretation. the presence of heterogeneity (p-value = 0.07). Although a broadly similar point estimate of d was produced by this random effects analysis, a much larger corresponding standard error of 0.15 was obtained. This is due to the large, but very imprecise, estimate of the between-study variance. This finding reinforces our statement that standard errors from our procedure will in general be too small if the common study effects model (Equation 1) assumes are not true and diagnostics, such as Figure 1, are essential to assess the evidence of any over-dispersion.
If sufficient data were available we would suggest pooling the estimates of d and f in a bivariate random effects meta-analysis at a second stage in a two stage approach. We leave the question concerning how much data we need, in order to favour a multivariate metaanalysis to the proposed univariate meta-regression based procedure, as an open question for further research. If there were very many studies providing information on the same age groups then we anticipate that multivariate random effects meta-analysis could prove useful for analysing heterogeneous data where the incidence rates, rather than the estimates of d and f, are the outcomes. In fact bivariate meta-analysis was used in the systematic review, to meta-analyse epidemiological studies that gave information by sex but whose published results did not offer rates stratified by age.
In some instances more sophisticated one-stage random effects modelling may be desirable and plots such as Figures 1-3 will be essential in order to determine whether this is needed. In situations where the observed rates appear to be more variable than predicted by the model, one straightforward way to proceed would be to incorporate a random effect into the resulting metaregressions in the usual way (Thompson and Sharp, 1999). However this standard methodology assumes that each study provides a single rate, so that the random effects are independent, but here it is plausible that any random effects within-studies may be correlated. For example, it could be that the unknown additional factors that result in betweenstudy variation for a particular study provide increased rates for older people, in which case the male and female random effects for particular age groups would be correlated. Since both the point estimates and the confidence intervals depend on both the modelling and the resulting estimates of the random effect structure, any particular one-stage random effects model fit, including those that model heterogeneity in the parameters of particular interest, could be hard to defend for the type of dataset that we have analysed.
Another pragmatic approach to dealing with any apparent over-dispersion is to use Thompson and Sharp's (1999) non-hierarchical based multiplicative over-dispersion parameter (their model 2) when fitting the meta-regressions, although Thompson and Sharp (1999) do not recommend this particular approach in practice when conducting more conventional metaregressions. This results in the same point estimates as our procedure but standard errors are not divided by the square root of mean square error. This increases the standard error of estimates of d and f by around 23% in our example, but statistically significant findings are still obtained. More sophisticated hierarchical modelling may be desirable in some instances but there are limits to what may be achieved when the available data comprise of just 50 rates from the available literature (Kirkbride et al., 2012). Rather than devoting any additional effort to the modelling of the random effects, it may instead be preferable to focus on the form of the regression in Equation 1, which assumes that rates vary continuously for men, but that there is a step change at SPO for women.
Quantifying and measuring the impact of the betweenstudy heterogeneity is often considered important in meta-analysis (Higgins and Thompson, 2002) and our approach somewhat complicates the assessment of this aspect of primary study data, because rather than providing a between-study variance parameter on which measures of heterogeneity (Higgins and Thompson, 2002) and inference (Biggerstaff and Jackson, 2008) can be based, our approaches uses a fixed effects approach. There is clear evidence that the estimated study specific effects a i differ in our sample but we have resisted the temptation to test for homogeneity, H 0 : a i = a, for all a i , as usually recommended (Higgins and Thompson, 2002).
In general we found between-study variation was very large in our systematic review; where standard random effects meta-analysis was used, I 2 statistics (Higgins and Thompson, 2002) in excess of 90% were not uncommon which we interpret as being indicative of very considerable heterogeneity. Partly because of these very large degrees of heterogeneity, and also because we were pooling rates from observational studies, when we presented pooled results from random effects meta-analyses in our systematic review we did so as descriptive rather than inferential statistics. Although we have performed formal inference for using our two parameters of interest here, we wish to make it clear that although our model adequately describes the data, all the usual cautions and caveats when interpreting statistical findings from observational studies should still be exercised here.
Our methodology was equally successful at detecting and describing the covariate effects of gender and SPO in women in many other study sets identified in the review, including for other epidemiological outcome measures.
By including these covariate effects of interest, in addition to the non-linear trend modelled by the fractional polynomial, our method provides a way of extending the type of approach introduced by Rota et al. (2010). Our methods made salient the features of this data that would be expected given existing results in the literature, while highlighting new features that were not necessarily obvious from simple visual inspection of the data (Table 2).
We used the midpoint of the age groups as covariates in our fractional polynomials and we recognize that the categorization of continuous variables can present genuine statistical issues . Furthermore we took the best fitting fractional polynomial and based all formal statistical inference on this, and we also recognize that this strategy is not without its limitations. Hence we provide the relevant data in Table 2 in the hope that it might continue to motivate methodological development on alternatives. R code is available from the first author on request to perform the type of analysis used and to facilitate the development of this type of statistical technique.