The use of percentage change from baseline as an outcome in a controlled trial is statistically inefficient: a simulation study. Typically a normal distribution is assumed for the outcome variable within each intervention group. ) Y Y Dependencies tend to be stronger if viewed over a wider range of values. odds ratio, risk ratio, hazard ratio, rate ratio). ∣ … In: Higgins JPT, Thomas J, Chandler J, Cumpston M, Li T, Page MJ, Welch VA (editors). σ Measurement scales are one particular type of ordinal outcome frequently used to measure conditions that are difficult to quantify, such as behaviour, depression and cognitive abilities. {\displaystyle x} ⁡ However, when used in a technical sense, correlation refers to any of several specific types of mathematical operations between the tested variables and their respective expected values. , the sample correlation coefficient can be used to estimate the population Pearson correlation It may be preferable, or necessary, to address the number of times these events occur rather than simply whether each person experienced an event or not (that is, rather than treating them as dichotomous data). {\displaystyle Y=X^{2}} 0 Y − ) In this circumstance it is necessary to standardize the results of the studies to a uniform scale before they can be combined. For further discussion of choice of effect measures for such sparse data (often with lots of zeros) see Chapter 10, Section 10.4.4. Y Calculations for the comparator group are performed in a similar way. Such results should be collected, as they may be included in meta-analyses, or – with certain assumptions – may be transformed back to the raw scale (Higgins et al 2008). This might be done either to improve interpretation of the results (see Chapter 15, Section 15.5), or because the majority of the studies present results after dichotomizing a continuous measure. {\displaystyle Y} In a sample of 1000 people, these numbers are 100 and 500 respectively. in a crossover trial, or simultaneous treatment of multiple sites on each individual); and. {\displaystyle Y} Williamson PR, Smith CT, Hutton JL, Marson AG. is a linear function of Charles Griffin & Co. pp 258–270. The types of outcome data that review authors are likely to encounter are dichotomous data, continuous data, ordinal data, count or rate data and time-to-event data. Again, if either of the SDs (at baseline and post-intervention) is unavailable, then one may be substituted by the other as long as it is reasonable to assume that the intervention does not alter the variability of the outcome measure. = The numbers 3.92, 3.29 and 5.15 are replaced with slightly larger numbers specific to the t distribution, which can be obtained from tables of the t distribution with degrees of freedom equal to the group sample size minus 1. Essentially, correlation is the measure of how two or more variables are related to one another. Karl Pearson developed the coefficient from a similar but slightly different idea by Francis Galton.[4]. ⁡ ( Difficulties will be encountered if studies have summarized their results using medians (see Section 6.5.2.5). For example, where early explanatory trials are combined with later pragmatic trials in the same review, pragmatic trials may include a wider range of participants and may consequently have higher SDs. ( X Numbers needed to treat are discussed in detail in Chapter 15, Section 15.4, as they are primarily used for the communication and interpretation of results. They also vary in the scale chosen to analyse the data (e.g. There are several different ways of comparing outcome data between two intervention groups (‘effect measures’) for each data type. {\displaystyle \sigma _{X}} It is common to use the term ‘event’ to describe whatever the outcome or state of interest is in the analysis of dichotomous data. , , However, means and medians can be very different from each other when the data are skewed, and medians often are reported because the data are skewed (see Chapter 10, Section 10.5.3). Down with odds ratios! Risk is the concept more familiar to health professionals and the general public. Meta-analysis of time-to-event data: a comparison of two-stage methods. The interpretation of the clinical importance of a given risk ratio cannot be made without knowledge of the typical risk of events without intervention: a risk ratio of 0.75 could correspond to a clinically important reduction in events from 80% to 60%, or a small, less clinically important reduction from 4% to 3%. ′ Methods (specifically polychotomous logistic regression models) are available for calculating study estimates of the log odds ratio and its SE. Unfortunately, it is not always clear which is being reported and some intelligent reasoning, and comparison with other studies, may be required. In the broadest sense correlation is any statistical association, though it commonly refers to the degree to which a pair of variables are linearly related. E X Collaboration with a knowledgeable statistician is advised if this approach is followed. A statistical confidence interval for true per cent reduction in caries-incidence studies. Deeks J. For interventions that reduce the chances of events, the odds ratio will be smaller than the risk ratio, so that, again, misinterpretation overestimates the effect of the intervention. = ¯ {\displaystyle X_{i}} Chapter 6: Choosing effect measures and computing estimates of effect. However, the causes underlying the correlation, if any, may be indirect and unknown, and high correlations also overlap with identity relations (tautologies), where no causal process exists. , If the majority of studies in a meta-analysis have missing SDs, these values should not be imputed. σ In informal parlance, correlation is synonymous with dependence. ) It may be difficult to identify the subset of participants who report both baseline and post-intervention measurements for whom change scores can be computed. For difference measures, a value of 0 represents no difference between the groups. ) = n , Vickers AJ. Zeros arise particularly when the event of interest is rare, such as unintended adverse outcomes. Y For example, eyes may be mistakenly used as the denominator without adjustment for the non-independence between eyes. Count data should not be treated as if they are dichotomous data (see Section 6.7). σ However, the clinical importance of a risk difference may depend on the underlying risk of events in the population. In contrast, Glass’ delta (Δ) uses only the SD from the comparator group, on the basis that if the experimental intervention affects between-person variation, then such an impact of the intervention should not influence the effect estimate. corr Consider the impact on the analysis of clustering, matching or other non- standard design features of the included studies. , A more detailed list of situations in which unit-of-analysis issues commonly arise follows, together with directions to relevant discussions elsewhere in this Handbook. For a ratio measure, such as a risk ratio, odds ratio or hazard ratio (which we denote generically as RR here), first calculate. Studies directly examining the correlation between financial status and quality and safety of patient care, however, have been equivocal and the findings uncertain. This is because the precision of a risk ratio estimate differs markedly between those situations where risks are low and those where risks are high. E The particular definition of SMD used in Cochrane Reviews is the effect size known in social science as Hedges’ (adjusted) g. This uses a pooled SD in the denominator, which is an estimate of the SD based on outcome data from both intervention groups, assuming that the SDs in the two groups are similar. {\displaystyle \rho _{X,Y}} 1 are jointly normal, uncorrelatedness is equivalent to independence. = Ranges are very unstable and, unlike other measures of variation, increase when the sample size increases. for Results from more than one time point for each study cannot be combined in a standard meta-analysis without a unit-of-analysis error. . X Alternatively, use can sometimes be made of aggregated data for each intervention group in each trial. Cochrane Handbook for Systematic Reviews of Interventions version 6.1 (updated September 2020). Relevant details of the t distribution are available as appendices of many statistical textbooks or from standard computer spreadsheet packages. Values higher and lower than these ‘null’ values may indicate either benefit or harm of an experimental intervention, depending both on how the interventions are ordered in the comparison (e.g. Sensitivity to the data distribution can be used to an advantage.  independent It is obtained by taking the ratio of the covariance of the two variables in question of our numerical dataset, normalized to the square root of their variances. Where exact P values are quoted alongside estimates of intervention effect, it is possible to derive SEs. BMC Medical Research Methodology 2012; 12: 9. Y Nevertheless, Hozo and colleagues conclude that the median may often be a reasonable substitute for a mean (Hozo et al 2005). ( As a general rule it is better to re-define such outcomes so that the analysis includes all randomized participants. Two unsatisfactory options are: (i) imputing zero functional ability scores for those who die (which may not appropriately represent the death state and will make the outcome severely skewed), and (ii) analysing the available data (which must be interpreted as a non-randomized comparison applicable only to survivors). Y {\displaystyle X} Johnston BC, Thorlund K, Schünemann HJ, Xie F, Murad MH, Montori VM, Guyatt GH. The examples are sometimes said to demonstrate that the Pearson correlation assumes that the data follow a normal distribution, but this is not correct.[4]. However, the method assumes that the differences in SDs among studies reflect differences in measurement scales and not real differences in variability among study populations. , and the conditional mean the risk ratio (RR; also called the relative risk); the risk difference (RD; also called the absolute risk reduction); and. Odds ratios, like odds, are more difficult to interpret (Sinclair and Bracken 1994, Sackett et al 1996). In contrast, switching the outcome can make a substantial difference for risk ratios, affecting the effect estimate, its statistical significance, and the consistency of intervention effects across studies. E given When either the baseline or post-intervention SD is unavailable, then it may be substituted by the other, providing it is reasonable to assume that the intervention does not alter the variability of the outcome measure. The correlation coefficient is symmetric: {\displaystyle Y} t For example, the Pearson correlation coefficient is defined in terms of moments, and hence will be undefined if the moments are undefined. ( ) Select a single time point and analyse only data at this time for studies in which it is presented. E corr {\displaystyle Y} Saving, process of setting aside a portion of current income for future use, or the flow of resources accumulated in this way over a given period of time. E These trials have similarities to crossover trials: whereas in crossover studies individuals receive multiple interventions at different times, in these trials they receive multiple interventions at different sites. y . Y ) Improving the interpretation of quality of life evidence in meta-analyses: the application of minimal important difference units. E The MD is required in the calculations from the t statistic or the P value. {\displaystyle Y} Y μ This requires the status of all patients in a study to be known at a fixed time point. is a widely used alternative notation for the correlation coefficient. are results of measurements that contain measurement error, the realistic limits on the correlation coefficient are not −1 to +1 but a smaller range. Odds can be converted to risks, and risks to odds, using the formulae: The interpretation of odds is more complicated than for a risk. Agresti A. Under this assumption, the statistical methods used for MDs would be used, with both the MD and its SE divided by the externally derived SD. What constitutes clinically important will depend on the outcome and the values and preferences of the person or population. We also use the term ‘risk ratio’ in preference to ‘relative risk’ for consistency with other terminology. A narrative approach might then be needed for the synthesis (see Chapter 12). To extract counts as time-to-event data, guidance in Section 6.8.2 should be followed. For example, a RoM might meaningfully be used to combine results from a study using a scale ranging from 0 to 10 with results from a study ranging from 1 to 50. X An alternative formula purely in terms of moments is, ρ The formula for converting an odds ratio to a risk ratio is provided in Chapter 15, Section 15.4.4. Guyot P, Ades AE, Ouwens MJ, Welton NJ. Systematic Reviews in Health Care: Meta-analysis in Context. Do you have feedback about the new online Handbook? If the sample size is small (say fewer than 60 participants in each group) then confidence intervals should have been calculated using a t distribution. Friedrich JO, Adhikari NK, Beyene J. Y Y {\displaystyle \operatorname {E} (Y)} X When there are more than two groups to combine, the simplest strategy is to apply the above formula sequentially (i.e. Risk describes the probability with which a health outcome will occur. X However, if we were to collect data only from 18- to 24-year-olds—represented by the shaded area of Figure 6.6—then the relationship would seem to be quite weak. Typically the natural log transformation (log base e, written ‘ln’) is used. Deeks JJ, Altman DG, Bradburn MJ. BMC Medical Research Methodology 2005; 5: 13. Furukawa and colleagues found that imputing SDs either from other studies in the same meta-analysis, or from studies in another meta-analysis, yielded approximately correct results in two case studies (Furukawa et al 2006). A general rule of thumb is to focus on the less common state as the event of interest. BMC Medical Research Methodology 2014; 14: 135. If Here we describe (1) how to calculate the correlation coefficient from a study that is reported in considerable detail and (2) how to impute a change-from-baseline SD in another study, making use of a calculated or imputed correlation coefficient. Define several different outcomes, based on different periods of follow-up, and plan separate analyses. {\displaystyle \operatorname {corr} (X,Y)=\operatorname {corr} (Y,X)} Sometimes the numbers of participants, means and SDs are not available, but an effect estimate such as a MD or SMD has been reported. They are known generically as survival data in the medical statistics literature, since death is often the event of interest, particularly in cancer and heart disease. Y j For example, when participants have particular symptoms at the start of the study the event of interest is usually recovery or cure. Measures of dependence based on quantiles are always defined. corr and {\displaystyle n\times n} { In research, risk is commonly expressed as a decimal number between 0 and 1, although it is occasionally converted into a percentage. In the example, these turn out to be. Does improved mood lead to improved health, or does good health lead to good mood, or both? The first step is to obtain the Z value corresponding to the reported P value from a table of the standard normal distribution. Absolute measures, such as the risk difference, are particularly useful when considering trade-offs between likely benefits and likely harms of an intervention. Continuous outcomes can be compared between intervention groups using a mean difference or a standardized mean difference. Sometimes it may be sensible to calculate the RR for more than one assumed comparator group risk. Equivalent expressions for Then A hazard ratio describes how many times more (or less) likely a participant is to suffer the event at a particular point in time if they receive the experimental rather than the comparator intervention. Select the longest follow-up from each study. {\displaystyle y} If some scales increase with disease severity (for example, a higher score indicates more severe depression) whilst others decrease (a higher score indicates less severe depression), it is essential to multiply the mean values from one set of studies by –1 (or alternatively to subtract the mean from the maximum possible value for the scale) to ensure that all the scales point in the same direction, before standardization. As a general rule, most methodologists believe that missing summary data (e.g. Different variations on the SMD are available depending on exactly what choice of SD is chosen for the denominator. Relevant details of the t distribution are available as appendices of many statistical textbooks or from standard computer spreadsheet packages. {\displaystyle \operatorname {E} (Y\mid X)} 1 An approximate SE of the log rate ratio is given by: A correction of 0.5 may be added to each count in the case of zero events. A log-rank analysis can be performed on these data, to provide the O–E and V values, although careful thought needs to be given to the handling of censored times. Y Sometimes the numbers of participants and numbers of events are not available, but an effect estimate such as an odds ratio or risk ratio may be reported. X Note that the methods in (2) are applicable both to correlation coefficients obtained using (1) and to correlation coefficients obtained in other ways (for example, by reasoned argument). X In these situations, and others where SEs cannot be computed, it is customary to add ½ to each cell of the 2✕2 table (for example, RevMan automatically makes this correction when necessary). The ‘odds’ refers to the ratio of the probability that a particular event will occur to the probability that it will not occur, and can be any number between zero and infinity. {\displaystyle X} Follmann D, Elliott P, Suh I, Cutler J. Variance imputation for overviews of clinical trials with continuous response. and/or Even though uncorrelated data does not necessarily imply independence, one can check if random variables are independent if their mutual information is 0. X Y s X Using the correlation coefficient calculated in step 1 above of 0.80, we can impute the change-from-baseline SD in the comparator group as: Missing mean values sometimes occur for continuous outcome data. Thus it is suitable for single (post-intervention) assessments but not for change-from-baseline measures (which can be negative). ( , determines this linear relationship: where , X Y 6.3 Extracting estimates of effect directly. Review authors should look for evidence of which one, and use a t distribution when in doubt. Specific considerations are required for continuous outcome data when extracting mean differences. Although it is preferable to decide how count data will be analysed in a review in advance, the choice often is determined by the format of the available data, and thus cannot be decided until the majority of studies have been reviewed. are the corrected sample standard deviations of X {\displaystyle s_{x}} ∈ , For 90% confidence intervals divide by 3.29 rather than 3.92; for 99% confidence intervals divide by 5.15. Correlation analysis is a statistical method used to evaluate the strength of relationship between two quantitative variables. {\displaystyle \operatorname {E} (X)} . Y {\displaystyle Y} Standard deviations can be obtained from a SE, confidence interval, t statistic or P value that relates to a difference between means in two groups (i.e. x {\displaystyle \rho _{X,Y}} Oxford (UK): Oxford University Press; 1990. , The Pearson correlation is defined only if both standard deviations are finite and positive. (See diagram above.) {\displaystyle Y} . {\displaystyle \operatorname {E} } Cluster-randomized studies, crossover studies, studies involving measurements on multiple body parts, and other designs need to be addressed specifically, since a naive analysis might underestimate or overestimate the precision of the study. Y 1 A Pearson product-moment correlation coefficient attempts to establish a line of best fit through a dataset of two variables by essentially laying out the expected values and the resulting Pearson's correlation coefficient indicates how far away the actual dataset is from the expected values. Consider the joint probability distribution of and Most of this chapter relates to this situation. σ Sometimes it is desirable to combine two reported subgroups into a single group. X For example, in an exchangeable correlation matrix, all pairs of variables are modeled as having the same correlation, so all non-diagonal elements of the matrix are equal to each other. Similarly, multiple treatment attempts per participant can cause a unit-of-analysis error. When effect measures are based on change from baseline, a single measurement is created for each participant, obtained either by subtracting the post-intervention measurement from the baseline measurement or by subtracting the baseline measurement from the post-intervention measurement. Measures of relative effect express the expected outcome in one group relative to that in the other. For example, a risk ratio of 3 for an intervention implies that events with intervention are three times more likely than events without intervention. ( For practical purposes, count data may be conveniently divided into counts of rare events and counts of common events. and , ∞ , denoted X Ades AE, Lu G, Dias S, Mayo-Wilson E, Kounali D. Simultaneous synthesis of treatment effects and mapping to a common scale: an alternative to standardisation. The two are interchangeable and both conveniently abbreviate to ‘RR’. {\displaystyle n} Bland derived an approximation for a missing mean using the sample size, the minimum and maximum values, the lower and upper quartile values, and the median (Bland 2015). The SD does not need to be modified. is completely determined by Clinically useful measures of effect in binary analyses of randomized trials. {\displaystyle \rho _{X,Y}} Illusory Correlation . Y , Review authors should approach multiple intervention groups in an appropriate way that avoids arbitrary omission of relevant groups and double-counting of participants (see MECIR Box 6.2.b) (see Chapter 23, Section 23.3). If, as the one variable increases, the other decreases, the rank correlation coefficients will be negative. usage the Newton's method for computing the nearest correlation matrix[18]) results obtained in the subsequent years. ⁡ and Y Editors: Julian PT Higgins, Tianjing Li, Jonathan J Deeks. The mean difference (MD, or more correctly, ‘difference in means’) is a standard statistic that measures the absolute difference between the mean value in two groups of a randomized trial. In all of these situations, a sensitivity analysis should be undertaken, trying different values of Corr, to determine whether the overall result of the analysis is robust to the use of imputed correlation coefficients. {\displaystyle {\overline {x}}} Analyses then proceed as for any other type of continuous outcome variable. Care must be taken to ensure that the number of participants randomized, and not the number of treatment attempts, is used to calculate confidence intervals. {\displaystyle y} = This is true of some correlation statistics as well as their population analogues. For example, means and SDs of logarithmic values may be available (or, equivalently, a geometric mean and its confidence interval). The true effects of interventions are never known with certainty, and can only be estimated by the studies available. The values of ratio measures of intervention effect (such as the odds ratio, risk ratio, rate ratio and hazard ratio) usually undergo log transformations before being analysed, and they may occasionally be referred to in terms of their log transformed values (e.g. This is because confidence intervals should have been computed using t distributions, especially when the sample sizes are small: see Section 6.5.2.3 for details. increases, and so does However, it is unlikely to be reasonable to combine RoM results from a study using a scale ranging from 0 to 10 with RoM results from a study using a scale ranging from 20 to 30: it is not possible to obtain RoM values outside of the range 0.67 to 1.5 in the latter study, whereas such values are readily obtained in the former study. The most commonly encountered effect measures used in randomized trials with dichotomous data are: Details of the calculations of the first three of these measures are given in Box 6.4.a. It may be impossible to pre-specify whether data extraction will involve calculation of numbers of participants above and below a defined threshold, or mean values and SDs. and Marinho VCC, Higgins JPT, Logan S, Sheiham A. Fluoride toothpaste for preventing dental caries in children and adolescents.
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