How to do meta analysis in stata forex
Often, the prognostic ability of a factor is evaluated in multiple studies. Two applications are presented. Introduction A prognostic factor is any measure that, among people with a given health condition, is associated with a subsequent clinical outcome 1 , 2. For example, in many cancers, tumour grade at the time of histological diagnosis is a prognostic factor because it is associated with time to disease recurrence or death; those with a higher tumour grade have a worse prognosis.
Prognostic factors thus distinguish groups of people with a different average prognosis, and this allows them to be useful for clinical practice and health research. For example, they can help define disease at diagnosis, inform clinical and therapeutic decisions either directly or as part of multivariable prognostic models , enhance the design and analysis of intervention trials and observational studies as they are potential confounders and may even identify targets for new interventions that aim to modify the course of a disease or health condition.
Given their importance, there are often hundreds of studies each year investigating the prognostic value of one or more bespoke factors in each disease field. In the course of contacting authors, additional data became available for two studies expanded since the original publication Burger et al, ; Andersson et al, Detailed information on the design of each of the 85 included studies is listed in the appendix.
Multiple infections 3. Cases with specimens considered to be inadequate for PCR testing were excluded. Type-specific prevalence is presented for the 18 most common HPV types as identified by this review HPV types 6, 16, 18, 31, 33, 35, 39, 45, 51, 52, 56, 58, 59, 66, 68, 70, 73 and 82 also known as MM4, W13B or IS39 in order of descending prevalence for each subgroup analysis.
For other consensus and type-specific PCR primers, only those HPV types specified in the individual reports were considered amplifiable. For HPV-specific prevalence, only studies testing for a particular HPV type contribute to the analysis for that type, and therefore sample size varies between the type-specific analyses. Statistical analyses Sources of variation in overall HPV prevalence were investigated by unconditional multiple logistic regression analysis Breslow and Day, Mean age and study year were found not to be significantly related to overall HPV prevalence.
Adjustment of overall HPV prevalence for these variables was done using the adjust command in Stata version 7. Confidence intervals for overall HPV prevalence were calculated assuming the nonindependence of cases within the same study using the cluster option in Stata White, Adjusted overall HPV prevalence varied between The most common HPV types identified were, in order of decreasing prevalence, HPV16, 18, 45, 31, 33, 58, 52, 35, 59, 56, 6, 51, 68, 39, 82, 73, 66 and Other HPV types were detected in no more than 0.
In cases from Africa, the prevalence of HPV45 8. In cases from Asia, HPV58 5. HPV18 was the predominant type However, more than 16 other HPV types were also associated with ICC, of which the most prevalent were types 45, 31, 33, 58 and 52 collectively accounting for However, it is not known to what extent other unknown sources of variation such as sample storage conditions, specific PCR conditions and quality of histopathology may affect these comparisons. Residual differences in prevalence between regions could also be because of the yet unknown HPV types not amplified by the existing PCR primers.
There were many similarities in HPV type-specific distribution across the regions studied. Other rarer types appeared to vary in their distribution. In most regions, HPV45, 31 and 33 were the third, fourth and fifth most common genotypes, although not necessarily in that order. Other types in SCC were too rare to make inferences on region-specific variations.
This intriguing finding does not appear to be because of differences with respect to region or HPV detection methods as it persisted even after adjusting for these factors. This difference has been described independently by many of the studies in this analysis and by studies outside the scope of this review IARC, Compared to HPV16, HPV18 has been shown to be associated with increasing oncogenic potential in cell culture Barbosa and Schlegel, , as well as a more rapid transition to malignancy Burger et al, and a poorer prognosis of cancer patients Nakagawa et al, ; Hildesheim et al, ; Schwartz et al, Given the fact that columnar tissue giving rise to ADC is less accessible, and possibly less susceptible to HPV infections, than the squamous tissue of SCC, the establishment of ADC may require a relatively more aggressive infection.
In addition to HPV16 and 18, this large meta-analysis facilitated the identification of differences for some rarer phylogenetically related types: the HPVrelated types 31, 33, 35, 52, and 58 were more prevalent in SCC All these differences were seen consistently in all regions where the comparison was possible. For all regions where histological comparison was possible, the ratio of ADC to SCC was higher than that reported by cancer registries Parkin et al, For example, ADC represent No material differences in results were observed when SCC was compared with cancers of unspecified histology.
Often, the prognostic ability of a factor is evaluated in multiple studies.
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| How to do meta analysis in stata forex | A clear connection between theory, data, and analysis is a hallmark of a great paper, reflected in that the more a meta-analysis attempts to test an existing theory, the larger the number of citations it receives Aguinis, Dalton, et al. For a more sophisticated option, there is the precision-effect test and a precision-effect estimate with standard errors PET-PEESEwhich can detect as well as correct for publication bias see Stanley and Doucouliagos for illustrative examples and code for Stata and SPSS. For example, in many cancers, tumour grade at the time of histological diagnosis is a prognostic factor because it is associated with check this out to disease recurrence or death; those with a higher tumour grade have a worse prognosis. Since we have established that the limitation of the existing software packages is handling descriptive data, we will be using rates in our example so that the difference in the final forest plot is more overt. In business, it is likely that influential reviews will increasingly become the purview of well-managed academic crowdsourcing projects i. |
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Once the missing data have been handled and all of the effect sizes have been converted into the same metric, we have our second dataset, which is ready for preliminary analysis. Different types of effect sizes There are many different types of effect sizes, some for continuous outcome variables and others for binary outcome variables.
The effect sizes for continuous outcome variables belong to one of two families of effect sizes: the d class and the r class. The d class effect sizes are usually calculated from the mean and standard deviation. They are a scaled difference between the means of two group. The r class effect sizes are also a ratio, but they are the ratio of variance attributable to an effect divided by the total effect, or more simply, the proportion of variance explained. Examples of this type of effect size include eta-squared and omega-squared.
These include the risk ratio, the odds ratio and the risk difference. The risk difference is an absolute measure, so it is very sensitive the number of baseline events. Converting between different types of effect sizes While you will need to collect information necessary to calculate an effect size from some articles, other articles will provide the effect size.
However, there are dozens of different types of effect sizes, so you may need to convert the effect size given in the paper into the type of effect size that you need for your meta-analysis. You need to be careful when using effect size converters, because some conversions make more sense than others.
Why is that? Bonnet and Kraemer have good summaries of issues regarding fourfold tables. Another point to keep in mind is the effect of rounding error when converting between different types of effect sizes. Like any other quantity calculated from sampled data, an effect size is an estimate. Because it is an estimate, we want to calculate the standard error or confidence interval around that estimate.
If you give the statistical software the information necessary to calculate the effect size, it will also calculate the standard error for that estimate. However, if you supply the estimate of the effect size, you will also need to supply either the standard error for the estimate or the confidence interval. This can be a real problem when an article reports an effect size but not its standard error, because it may be difficult to find a way to derive that information from what is given in the article.
Despite the large number of effect sizes available, there are still some situations in which there is no agreed-upon measure of effect. Two examples are count models and multilevel models. Data inspection and descriptive statistics In the preliminary analysis, we do all of the data checking that you would do with any other dataset, such as look for errors in the data, get to know the variables, etc. Of particular interest is looking at the estimates of the effect sizes for outliers.
Of course, what counts as an outlier depends on the context, but you still want to identify any extreme effect sizes. If the dataset is small, you can simply look through the data, but if the dataset is large, you may need to use a test to identify outliers. You could use the chi-square test for outliers, the Dixon Q test , for outliers or the Grubbs , , test for outliers.
We used a method proposed by Viechtbauer and Cheung, , and there are others that you could use as well. The other problem was that if one outlier was removed and the test rerun, a different point would be identified as an outlier. After you have done the descriptive statistics and considered potential outliers, it is finally time to do the analysis!
Before running the meta-analysis, we should discuss two important topics that will be shown in the output. The first is weighting, and the second is measures of heterogeneity. Weighting As we know, some of the studies had more subjects than others.
In general, the larger the N, the lower the sampling variability and hence the more precise the estimate. Because of this, the studies with larger Ns are given more weight in a meta-analysis than studies with smaller Ns. These weights are relative weights and should sum to You do not need to calculate these weights yourself; rather, the software will calculate and use them, and they will be shown in the output.
This variability is actually comprised of two components: the variation in the true effect sizes, which is called heterogeneity, and spurious variability, which is just random error i. When conducting a meta-analysis, we want to get a measure of this heterogeneity, or the variation in the true effect sizes.
There are several measures of this, and we will discuss each in turn. The explanations found there include useful graphs; reading that chapter is highly recommended. If the heterogeneity was in fact 0, it would mean that all of the studies in the meta-analysis shared the same true effect size. However, we would not expect all of the effect sizes to be the exact same value, because there would with-study sampling error.
Instead, the effect sizes would fall within a particular range around the true effect. Now suppose that the true effect size does vary between studies. In this scenario, the observed effect sizes vary for two reasons: Heterogeneity with respect to the true effect sizes and within-study sampling error. Now we need to separate the heterogeneity from the within-study sampling error. Looking back at the three steps listed above, the first step is to calculate Q.
There is a formula for doing this by hand, but most researchers use a computer program to do this. Once you have Q, the next step is to calculate the expected value of Q, assuming that all studies share a common effect size and hence all of the variation is due to sampling error within studies.
If you want to know if the heterogeneity is statistically significant, you can do so with Q and df. Specifically, the null hypothesis is that all studies share a common effect size, and under this null hypothesis, Q will follow a central chi-squared distribution with degrees of freedom equal to k — 1. As you would expect, this test is sensitive to both the magnitude of the effect i.
While a statistically-significant p-value is evidence that the true effects vary, the converse is not true. In other words, you should not interpret a non-significant result. The result could be non-significant because the true effects do not vary, or because there is not enough power to detect the effect, or some other reason. This has all been pretty easy to calculate, but there are some limitations to Q. First of all, the metric is not intuitive.
Also, Q is a sum, not a mean, which means that it is very sensitive the number of studies included in the meta-analysis. But calculating Q has not been a waste of time, because it is used in the calculation of other measures of heterogeneity that may be more useful. If we take Q, remove the dependence on the number of studies and return it to the original metric, then we have T2, which is an estimate of variance of the true effects. If we take Q, remove the dependence on the number of studies and express the result as a ratio, we will have I2, which estimates the proportion of the observed variance that is heterogeneity as opposed to random error.
Tau-squared is defined as the variance of the true effect sizes. To know this, we would need to have an infinite number of studies in our meta-analysis, and each of those studies would need to have an infinite number of subjects.
Rather, we can estimate tau-squared by calculating T2. To do this, we start with Q — df and divide this quantity by C. If tau-squared is the actual value of the variance and T2 is the estimate of that actual value, then you can probably guess that tau is the actual standard deviation and T is the estimate of this parameter.
While tau-squared can never be less than 0 because the actual variance of the true effects cannot be less than 0 , T2 can be less than 0 if the observed variance is less than expected based on the within-study variance i. When this happens, T2 should be set to 0. I2 Notice that T2 and T are absolute measures, meaning that they quantify deviation on the same scale as the effect size index. I2 can be thought of as a type of signal-to-noise ratio. I2 is a descriptive statistic and not an estimate of any underlying quantity.
Borenstein, et. As such it is convenient to view I2 as a measure of inconsistency across the findings of the studies, and not as a measure of the real variation across the underlying true effects. An I2 value near 0 means that most of the observed variance is random; it does not mean that the effects are clustered in a narrow range.
For example, the observed effects could vary widely because the studies had a lot of sampling error. Instead, they could have a very narrow range and be estimated with great precision. The point here is to stress that I2 is a measure of proportion of variability, not a measure of the amount of true variability.
There are several advantages to using I2. It can be interpreted as a ratio, similar to indices used in regression and psychometrics. Finally, I2 is not directly influenced by the number of studies included in the meta-analysis. Because I2 is on a relative scale, you should look at it first to decide if there is enough variation to warrant speculation about the source or cause of the variation.
In other words, before jumping into a meta-regression or subgroup analysis, you want to look at I2. If it is really low, then there is no point to doing a meta-regression or subgroup analysis. The N, mean and standard deviation for the experimental group is given first, and then those values for the control group.
One option on the —metan- command was used, and that was the —hedges- option. We used it because some of the studies included in our meta-analysis had fairly low Ns, and because there is no penalty for using it if it is not needed. By default, the —metan- command produces two types of output: the table of results and a forest plot. Notice that in this example, all but two of the confidence intervals include 0.
At the bottom of the table, we see the mean effect size, which is 0. Notice that this confidence interval does not include 0. This means that, based on this meta-analysis, the effect size is 0. The next line of the output below the table give estimates of the heterogeneity. The Heterogeneity chi-squared equals 5. Congruent with that is the estimate of I-squared, which is 0. Having an I-squared equal to 0 is somewhat unusual, and it means that there is no point to running any meta-regressions which are used to explain heterogeneity.
The summary effect of 0. For each study, a square shows its place on the scale and the confidence interval is represented by the line on either side of the square. For Study 6, there is an arrow on the right side of the confidence interval, which indicates that the confidence interval is wider on that side than the highest value on the scale but that is difficult to see because of the rounded of the values for the CIs.
Lots of options can be added to make the forest plot ready for publication, and almost all published meta-analyses include a forest plot. In fact the use of the term fixed effect in connection with meta-analysis is at odds with the usual meaning of fixed effects in statistics. A more suitable term for the fixed-effect meta-analysis might be a common-effect meta-analysis. The term fixed effects is traditionally used in another context with a different meaning.
Concretely, we can talk about the subgroups as being fixed in the sense of fixed rather than random. This has important implications when talking about subgroup analyses, because in that context, a mixed effect meta-analysis means a fixed effect subgroup across groups while a random-effects model was used for the within-group analysis. Fixed effect models and random effect models make different assumptions, and you should choose between these options based on your assessment of how well your data meet the assumptions of these types of models.
Fixed effect models are appropriate if two conditions are satisfied. The first is that all of the studies included in the meta-analysis are identical in all important aspects. Secondly, the purpose of the analysis is to compute the effect size for a given population, not to generalize the results to other populations.
How reasonable do they seem to you? On the other hand, you may think that because the research studies were conducted by independent researchers, there is no reason to believe that the studies are functionally equivalent. Also, given the differences between the studies, you might want to generalize your results to a range of similar but not identical situations or scenarios. In an ideal world, you choose between a fixed and random effects meta-analysis based on the assumptions that you were willing to make.
In reality, though, other considerations are also important. One of those considerations is the size of your dataset. Just as a study with a small N is unlikely to capture the true amount of variability in the population, so a meta-analysis with few studies is likely to produce a precise estimate of the between-studies variance. This means that if even if we preferred the random-effect model, our dataset may not contain the necessary or amount of information.
You have some options in this situation, but each option comes with a down side. One option is to report the effect sizes for each study but omit the summary effect size. The down side that is that some readers and possibly journal reviewers will not understand that conclusions should not be drawn from the summary effect and its confidence interval and wonder why it has been omitted.
Another possible down side is that readers may generalize your results, even if you state that such a generalization is not warranted. A third, and possibly the best option, is to run the analysis as a Bayesian meta-analysis. In this approach, the estimate of tau-squared is based on data from beyond the studies included in the current analysis. Here is the syntax for a random-effects meta-analysis using our example data. However, some may be unobtainable because of publication bias.
Publication bias is the bias by publishers of academic journals to prefer to publish studies reporting statistically significant results rather than studies reporting statistically non-significant results. In a similar vein, researchers may be loath to write up a paper reporting statistically non-significant results on the belief that the paper is more likely to be rejected. The effect on a meta-analysis is that there could be missing data i. This, of course, leads to a biased estimate of the summary effect.
One other point to keep in mind: For any given sample size, the result is more likely to be statistically significant if the effect size is large. Hence, publication bias refers to both statistically significant results and large effect sizes.
There are other types of bias that should also be considered. Availability bias: including those studies that are easiest for the meta-analyst to access To which journals does your university subscribe? Cost bias: including those studies that are freely available or lowest cost To which journals does your university subscribe?
They were proposed by Begg, et. However, both of these test suffer from several limitations. First, the tests and the funnel plot itself may yield different results simply by changing the metric of the effect size. Second, both a reasonable number of studies must be included in the analysis, and those studies must have a reasonable amount of dispersion. Finally, these tests are often under-powered; therefore, a non-significant result does not necessarily mean that there is no bias.
Tests and graphs to detect publication bias The meta bias command can be used to assess bias. The effect size and its standard error are provided, and there are three options that can be used. They are egger, harbord and peters. The harbord and peters options can only be used with binary data. In the example below, the egger option is used. Remember that the effect size is usually on the x-axis and the sample size or variance on the y-axis with the largest sample size or smallest variance at the top.
If there is no publication bias, then the studies will be distributed evenly around the mean effect size. Smaller studies will appear near the bottom because they will have more variance than the larger studies which are at the top of the graph. If there is publication bias, then there will seem to be a few studies missing from the middle left of the graph, and very few, if any, studies in the lower left of the graph.
The lower left being where small studies reporting small effect sizes would be. The numbers in parentheses give the range of p-values. In the presence of publication bias, this summary effect would be larger than it should be. If the missing studies were included in the analysis with no publication bias , the summary effect might no longer be statistically significant.
If only a few studies were needed to render our statistically significant summary effect non-significant, then we should be quite worried about our observed result. First, it focuses on statistical significance rather than practical, or real world, significance. As we have seen, there can be quite a difference between these two. Second, it assumes that the mean of the missing effect sizes is 0, but it could negative or slightly positive.
If it was negative, then fewer studies would be needed to render our summary effect non-significant. To understand how this is done, think of a funnel plot that shows publication bias, meaning that there are studies in the lower right of the plot but few, if any, on the lower left.
The advantages to this approach are that it gives an estimate of the unbiased effect size, and there is usually a graph associated with it that is easy to understand it usually includes the imputed studies. The disadvantages include a strong assumption about why the missing studies are missing and that one or two really aberrant studies can have substantial influence on the results. The meta trimfill command can be used. The same is true for the creation of the funnel plot.
The first line in the table of the cumulative meta-analysis shows the summary effect based on only the first study. The second line in the table shows the summary effect based on only the first two studies, and so on. Of course, the final summary effect will be the same as from the regular meta-analysis, because both are based on all of the studies. The studies can be sorted in different ways to address different questions. For example, if you want to look at only the largest studies to see when the estimate of the summary effect size stabilizes, you sort the studies based on N.
Or you might be interested in sorting the studies by year of publication. It is really important to know if this surgical technique increases life expectancy, so many studies are done. The question is, at what point in time have enough studies been done to answer this question? A third use for a cumulative meta-analysis is as a method to detect publication bias. For this, you would sort the studies from most to least precise.
You might suspect publication bias if the effects in the most precise studies were small but increased as the less precise studies were added. The forest plot would show not only whether there was a shift, but also the magnitude of the shift. Cumulative meta-analysis can be done with the meta summarize command with the cumulative option. In the example below, the random option was used to specify a random-effects meta-analysis. In the parentheses after the cumulative option is the name of the variable by which the meta-analysis is cumulative.
In this example, it is the publication date. They were very careful to point out that there is no way to know why this is true. It could be publication bias, or it could be that the smaller studies, especially if they were the first studies done, included subjects who were more ill, more motivated, more something, than the later-conducted studies that included more subjects.
It is also possible that the smaller studies had better quality control. In the end, any one of these reasons, other reasons, or any combination thereof may explain why the smaller studies reported larger effects. This is important to remember when writing up results.
Up until now, we have assumed that the variability was caused by random error sampling error in the individual studies or some other, as yet undiscussed, source of variability. These other sources of variability could be that some studies in the meta-analysis compared drug A with dosage X to placebo, while the other studies compared drug A with dosage 2X to placebo. Or perhaps some of the studies included only females, while the rest included only males.
This is called a subgroup analysis. In meta-analysis, you do something similar, but you just have more options of types of models. One option is to use a fixed-effects model. A second option is to use a random-effects model using an estimate of tau-squared from each group.
A third option is to use a random-effects model using a pooled estimate of tau-squared. In addition to these options, you also need to choose your method: a Z-test, a Q-test based on an analysis of variance, or a Q-test for heterogeneity. Any of the methods can be used with any of the models, creating nine different possibilities.
Your decision regarding choice of model should be based on what you know about your dataset. Your choice of method should be based on the type of information you are seeking about your data and the type of conclusion you wish to draw. Another decision you need to make is if a summary effect from all groups combined should be reported, or if only the summary effects for each group should be reported.
Again, this will depend on your data and your purpose. Please see Chapter 19 in Borenstein, et. When analyzing primary data, a regression model is used when there are one or more predictors to be associated with an outcome. With a meta-regression, the predictors are at the level of the study, and the outcome is the effect size. The purpose of the meta-regression is to explain the heterogeneity.
Of course, this presumes that there is any heterogeneity to be explained. Also, you need to consider the size of your meta-analysis dataset. In my dataset, there were only eleven studies, so even if there was some heterogeneity to be explained, I would have had difficulty running a meta-regression, and at most, only one predictor could be included. If there is heterogeneity to be explained and the dataset is large enough, though, almost all of the regression techniques are available, including the use of interaction terms, quadratic terms, logistic regression, etc.
Of course, the limitations encountered when analyzing primary data are also found with meta-regressions. For example, alpha inflation from multiple comparisons , power issues, etc. In Summary measures and homogeneity test, we established the presence of heterogeneity between the study results. As we said in Example dataset: Effects of teacher expectancy on pupil IQ, it was suspected that the amount of contact between the teachers and students before the experiment may explain some of the between-study variability.
Below Meta-regression, we explore the impact of continuous weeks on the effect sizes. For categorical variables, we can perform subgroup analysis—separate meta-analysis for each group—to explore heterogeneity between the groups. Mean Diff. Effect size: stdmdiff Std. In our example, we specified only one grouping variable, week1, but you can include more, provided you have a sufficient number of studies per group.
The test of no differences between the groups, reported at the bottom of the graph, is rejected with a chi-squared test statistic of Meta-regression Meta-regression is often used to explore heterogeneity induced by the relationship between moderators and study effect sizes.
Moderators may include a mixture of continuous and categorical variables. In Stata, you perform meta-regression by using meta regress. Continuing with our heterogeneity analysis, let's use meta-regression to explore the relationship between study-specific effect sizes and the amount of prior teacher—student contact weeks. Interval] weeks -. Postestimation: Bubble plot Continuing with Meta-regression, we can produce a bubble plot after meta-regression with one continuous covariate to explore the relationship between the effect sizes and the covariate.
There are also several outlying studies in the region where weeks is less than roughly 3 weeks.
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