Progressive statistics.
Abstract: Statistical guidelines and expert statements are now available to assist in the analysis and reporting of studies in some biomedical disciplines. We present here a more progressive resource for sample-based studies, meta-analyses and case studies in sports medicine and exercise science. We offer forthright advice on the following controversial or novel issues: using precision of estimation for inferences about population effects in preference to null-hypothesis testing, which is inadequate for assessing clinical or practical importance; justifying sample size via acceptable precision or confidence for clinical decisions rather than via adequate power for statistical significance; showing standard deviations rather than standard errors of the mean, to better communicate magnitude of differences in means and non-uniformity of error; avoiding purely non-parametric analyses, which cannot provide inferences about magnitude and are unnecessary; using regression statistics in validity studies, in preference to the impractical and biased limits of agreement; making greater use of qualitative methods to enrich sample-based quantitative projects; and seeking ethics approval for public access to the depersonalized raw data of a study, to address the need for more scrutiny of research and better meta-analyses. Advice on less contentious issues includes: using covariates in linear models to adjust for confounders, to account for individual differences, and to identify potential mechanisms of an effect; using log transformation to deal with non-uniformity of effects and error; identifying and deleting outliers; presenting descriptive, effect and inferential statistics in appropriate formats; and contending with bias arising from problems with sampling, assignment, blinding, measurement error, and researchers' prejudices. This article should advance the field by stimulating debate, promoting innovative approaches, and serving as a useful checklist for authors, reviewers and editors. KEYWORDS: analysis, case, design, inference, qualitative, quantitative, sample
Article Type: Report
Subject: Mathematical statistics (Usage)
Research (Methods)
Authors: Hopkins, Will G.
Batterham, Alan M.
Marshall, Stephen W.
Hanin, Juri
Pub Date: 01/01/2009
Publication: Name: Sportscience Publisher: Internet Society for Sport Science Audience: Academic Format: Magazine/Journal Subject: Health Copyright: COPYRIGHT 2009 Internet Society for Sport Science ISSN: 1174-9210
Issue: Date: Annual, 2009 Source Volume: 13
Product: Product Code: 8524300 Statistics; 8515100 Basic Research; 8500000 Science, Research & Development NAICS Code: 54171 Research and Development in the Physical, Engineering, and Life Sciences; 5417 Scientific Research and Development Services
Geographic: Geographic Scope: United States Geographic Code: 1USA United States
Accession Number: 297310266
Full Text: In response to the widespread misuse of statistics in research, several biomedical organizations have published statistical guidelines in their journals, including the International Committee of Medical Journal Editors (, the American Psychological Association (Anonymous, 2001), and the American Physiological Society (Curran-Everett and Benos, 2004). Expert groups have also produced statements about how to publish reports of various kinds of medical research (Table 1). Some medical journals now include links to these statements as part of their instructions to authors.

In this article we provide our view of best practice for the use of statistics in sports medicine and the exercise sciences. The article is similar to those referenced in Table 1 but includes more practical and original material. It should achieve three useful outcomes. First, it should stimulate interest and debate about constructive change in the use of statistics in our disciplines. Secondly, it should help legitimize the innovative or controversial approaches that we and others sometimes have difficulty including in publications. Finally, it should serve as a statistical checklist for researchers, reviewers and editors at the various stages of the research process. Not surprisingly, some of the reviewers of this article disagreed with some of our advice, so we emphasize here that the article represents neither a general consensus amongst experts nor editorial policy for this journal. Indeed, some of our innovations may take decades to become mainstream.

Most of this article is devoted to advice on the various kinds of sample-based studies that comprise the bulk of research in our disciplines. Table 2 and the accompanying notes deal with issues common to all such studies, arranged in the order that the issues arise in a manuscript. This table applies not only to the usual studies of samples of individuals but also to meta-analyses (in which the sample consists of various studies) and quantitative non-clinical case studies (in which the sample consists of repeated observations on one subject). Table 3, which should be used in conjunction with Table 2, deals with additional advice specific to each kind of sample-based study and with clinical and qualitative single-case studies. The sample-based studies in this table are arranged in the approximate descending order of quality of evidence they provide for causality in the relationship between a predictor and dependent variable, followed by the various kinds of methods studies, meta-analyses, and the single-case studies. For more on causality and other issues in choice of design for a study, see Hopkins (2008).

Note 1: Inferences

Inferences are evidence-based conclusions about the true nature of something. The traditional approach to inferences in research on samples is an assertion about whether the effect is statistically significant or "real", based on a P value. Specifically, when the range of uncertainty in the true value of an effect represented by the 95% confidence interval does not include the zero or null value, P is <0.05, the effect "can't be zero", so the null hypothesis is rejected and the effect is termed significant; otherwise P is >0.05 and the effect is non-significant. A fundamental theoretical dilemma with this approach is the fact that the null hypothesis is always false; indeed, with a large enough sample size all effects are statistically significant. On a more practical level, the failure of this approach to deal adequately with the real-world importance of an effect is evident in the frequent misinterpretation of a non-significant effect as a null or trivial effect, even when it is likely to be substantial. A significant effect that is likely to be trivial is also often misinterpreted as substantial.

A more realistic and intuitive approach to inferences is based on where the confidence interval lies in relation to threshold values for substantial effects rather than the null value (Batterham and Hopkins, 2006). If the confidence interval includes values that are substantial in some positive and negative sense, such as beneficial and harmful, you state in plain language that the effect could be substantially positive and negative, or more simply that the effect is unclear. Any other disposition of the confidence interval relative to the thresholds represents a clear outcome that can be reported as trivial, positive or negative, depending on the observed value of the effect. Such magnitude-based inferences about effects can be made more accurate and informative by qualifying them with probabilities that reflect the uncertainty in the true value: possibly harmful, very likely substantially positive, and so on. Note that even when an effect is unclear, you can often make a useful probabilistic statement about it (e.g., probably beneficial), and your findings should contribute to a meta-analysis. The qualitative probabilistic terms can be assigned using the following scale (Hopkins, 2007): <0.5%, most unlikely, almost certainly not; 0.5-5%, very unlikely; 5-25%, unlikely, probably not; 25-75%, possibly; 75-95%, likely, probably; 95-99.5%, very likely; >99.5%, most likely, almost certainly. Research on the perception of probability could result in small adjustments to this scale.

Use of thresholds for moderate and large effects allows even more informative inferential assertions about magnitude, such as probably moderately positive, possibly associated with small increase in risk, almost certain large gain, and so on. Uncertainty in magnitude can also be indicated by stating the magnitudes of the lower and upper confidence limits as a range (e.g., trivial-moderate benefit). Some effect statistics have generally accepted magnitude thresholds for such assertions. Thresholds of 0.1, 0.3 and 0.5 for small, moderate and large correlation coefficients suggested by Cohen (1988) can be augmented with 0.7 and 0.9 for very large and extremely large; these translate approximately into 0.20, 0.60, 1.20, 2.0 and 4.0 for standardized differences in means (the mean difference divided by the appropriate between-subject SD) and into maximum risk differences of 10%, 30%, 50%, 70% and 90% (see new The maximum risk differences between two groups translate approximately into hazard ratios of 1.3, 2.3, 4.5, 10 and 100 for a common injury, illness or other event (one that most individuals will experience eventually); standardization of differences in means of the log of time to the event in the two groups produces similar thresholds for the hazard ratio. For less common or rare events, hazard-ratio thresholds of 1.11, 1.4, 2.0, 3.3 and 10 are justifiable on the grounds that the corresponding proportions of cases attributable to the exposure or effect under investigation are 10%, 30%, 50%, 70% and 90% (Hopkins, 2009). These proportions, interpreted as an extra medal on average in 1, 3, 5, 7 and 9 competitions per 10 competitions, provide thresholds for change in a top athlete's competition time or distance of 0.3, 0.9, 1.6, 2.5 and 4.0 of the within-athlete variation between competitions (Hopkins et al., 1999 and WGH, unpublished observations). Thresholds have been suggested for some diagnostic statistics (Jaeschke et al., 1994), but more research is needed on these and on thresholds for the more usual measures of validity and reliability.

An appropriate default level of confidence for the confidence interval is 90%, because it implies quite reasonably that an outcome is clear if the true value is very unlikely to be substantial in a positive and/or negative sense. Use of 90% rather than 95% has also been advocated as a way of discouraging readers from reinterpreting the outcome as significant or non-significant at the 5% level (Sterne and Smith, 2001). In any case, a symmetrical confidence interval of whatever level is appropriate for making only non-clinical or mechanistic inferences. An inference or decision about clinical or practical utility should be based on probabilities of harm and benefit that reflect the greater importance of avoiding use of a harmful effect than failing to use a beneficial effect. Suggested default probabilities for declaring an effect clinically beneficial are <0.5% (most unlikely) for harm and >25% (possible) for benefit (Hopkins, 2007). A clinically unclear effect is therefore possibly beneficial (>25%) with an unacceptable risk of harm (>0.5%). Equivalently, an unclear effect occurs when an asymmetric confidence interval that is a 99% interval on the harmful side of an observed effect and a 50% interval on the beneficial side overlaps into harmful and beneficial values. (The disposition of an asymmetric confidence interval also underlies the appropriate interpretation of statistical significance.) The probabilities of >25% for benefit and <0.5% for harm correspond to a minimum ratio of 66 for odds of benefit to odds of harm, a suggested default when sample sizes are sub-or supra-optimal (Hopkins, 2007). Thus you could decide to make use of an effect with an 80% chance of benefit and a 5% chance of harm, because the odds of benefit outweigh the odds of harm by a factor of 76, which is >66.

Magnitude-based inferences as outlined above represent a subset of the kinds of inference that are possible using so-called Bayesian statistics, in which the researcher combines the study outcome with uncertainty in the effect prior to the study to get the posterior (updated) uncertainty in the effect. A qualitative version of this approach is an implicit and important part of the Discussion section of most studies, but in our view specification of the prior uncertainty is too subjective to apply the approach quantitatively. Researchers may also have difficulty accessing and using the computational procedures. On the other hand, confidence limits and probabilities related to threshold magnitudes can be derived readily via a spreadsheet (Hopkins, 2007) by making the same assumptions about sampling distributions that statistical packages use to derive P values. Bootstrapping, in which a sampling distribution for an effect is derived by resampling from the original sample thousands of times, also provides a robust approach to computing confidence limits and magnitude-based probabilities when data or modeling are too complex to derive a sampling distribution analytically.

Note 2: Access to Data

Public access to depersonalized data, when feasible, serves the needs of the wider community by allowing more thorough scrutiny of data than that afforded by peer review and by leading to better meta-analyses. Make this statement in your initial application for ethics approval, and state that the data will be available indefinitely at a website or on request without compromising the subjects' privacy.

Note 3: Multiple Inferences

Any conclusive inference about an effect could be wrong, and the more effects you investigate, the greater the chance of making an error. If you test multiple hypotheses, there is inflation of the Type I error rate: an increase in the chance that a null effect will turn up statistically significant. The usual remedy of making the tests more conservative is not appropriate for the most important pre-planned effect, it is seldom applied consistently to all other effects reported in a paper, and it creates problems for meta-analysts and other readers who want to assess effects in isolation. We therefore concur with others (e.g., Perneger, 1998) who advise against adjusting the Type I error rate or confidence level of confidence intervals for multiple effects.

For several important clinical or practical effects, you should nevertheless constrain the increase in the chances of making clinical errors. Overall chances of benefit and harm for several interdependent effects can be estimated properly by bootstrapping, but a more practical and conservative approach is to assume the effects are independent and to estimate errors approximately by addition. The sum of the chances of harm of all the effects that separately are clinically useful should not exceed 0.5% (or your chosen maximum rate for Type 1 clinical errors-see Note 4); otherwise you should declare fewer effects useful and acknowledge that your study is underpowered. Your study is also underpowered if the sum of chances of benefit of all effects that separately are not clinically useful exceeds 25% (or your chosen Type 2 clinical error rate). When your sample size is small, reduce the chance that the study will be underpowered by designing and analyzing it for fewer effects.

A problem with inferences about several effects with overlapping confidence intervals is misidentification of the largest (or smallest) and upward (or downward) bias in its magnitude. In simulations the bias is of the order of the average standard error of the outcome statistic, which is approximately one-third the width of the average 90% confidence interval (WGH, unpublished observations). Acknowledge such bias when your aim is to quantify the largest or smallest of several effects.

Note 4: Sample Size

Sample sizes that give acceptable precision with 90% confidence limits are similar to those based on a Type 1 clinical error of 0.5% (the chance of using an effect that is harmful) and a Type 2 clinical error of 25% (the chance of not using an effect that is beneficial). The sample sizes are approximately one-third those based on the traditional approach of an 80% chance of statistical significance at the 5% level when the true effect has the smallest important value. Until hypothesis testing loses respectability, you should include the traditional and new approaches in applications for ethical approval and funding.

Whatever approach you use, sample size needs to be quadrupled to adequately estimate individual differences or responses and effects of covariates on the main effect. Larger samples are also needed to keep clinical error rates for clinical or practical decisions acceptable when there is more than one important effect in a study (Note 3). See Reference (Hopkins, 2006a) for a spreadsheet and details of these and many other sample-size issues.

Note 5: Mechanisms

In a mechanisms analysis, you determine the extent to which a putative mechanism variable mediates an effect through being in a causal chain linking the predictor to the dependent variable of the effect. For an effect derived from a linear model, the contribution of the mechanism (or mediator) variable is represented by the reduction in the effect when the variable is included in the model as another predictor. Any such reduction is a necessary but not sufficient condition for the variable to contribute to the mechanism of the effect, because a causal role can be established definitively only in a separate controlled trial designed for that purpose.

For interventions, you can also examine a plot of change scores of the dependent variable vs those of potential mediators, but beware that a relationship will not be obvious in the scatter-gram if individual responses are small relative to measurement error. Mechanism variables are particularly useful in unblinded interventions, because evidence of a mechanism that cannot arise from expectation (placebo or nocebo) effects is also evidence that at least part of the effect of the intervention is not due to such effects.

Note 6: Linear Models

An effect statistic is derived from a model (equation) linking a dependent (the "Y" variable) to a predictor and usually other predictors (the "X" variables or covariates). The model is linear if the dependent can be expressed as a sum of terms, each term being a coefficient times a predictor or a product of predictors (interactions, including polynomials), plus one or more terms for random errors. The effect statistic is the predictor's coefficient or some derived form of it. It follows from the additive nature of such models that the value of the effect statistic is formally equivalent to the value expected when the other predictors in the model are held constant. Linear models therefore automatically provide adjustment for potential confounders and estimates of the effect of potential mechanism variables. A variable that covaries with a predictor and dependent variable is a confounder if it causes some of the covariance and is a mechanism if it mediates it. The reduction of an effect when such a variable is included in a linear model is the contribution of the variable to the effect, and the remaining effect is independent of (adjusted for) the variable.

The usual models are linear and include: regression, ANOVA, general linear and mixed for a continuous dependent; logistic regression, Poisson regression, negative binomial regression and generalized linear modeling for events (a dichotomous or count dependent); and proportional-hazards regression for a time-to-event dependent. Special linear models include factor analysis and structural equation modeling.

For repeated measures or other clustering of observations of a continuous dependent variable, avoid the problem of interdependence of observations by using within-subject modeling, in which you combine each subject's repeated measurements into a single measure (unit of analysis) for subsequent modeling; alternatively, account for the interdependence using the more powerful approach of mixed (multilevel or hierarchical) modeling, in which you estimate different random effects or errors within and between clusters. Avoid repeated-measures ANOVA, which sometimes fails to account properly for different errors. For clustered event-type dependents (proportions or counts), use generalized estimation equations.

Note 7: Non-parametric Analysis

A requirement for deriving inferential statistics with the family of general linear models is normality of the sampling distribution of the outcome statistic. Although there is no test that data meet this requirement, the central-limit theorem ensures that the sampling distribution is close enough to normal for accurate inferences, even when sample sizes are small (~10) and especially after a transformation that reduces any marked skewness in the dependent variable or non-uniformity of error. Testing for normality of the dependent variable and any related decision to use purely non-parametric analyses (which are based on rank transformation and do not use linear or other parametric models) are therefore misguided. Such analyses lack power for small sample sizes, do not permit adjustment for covariates, and do not permit inferences about magnitude. Rank transformation followed by parametric analysis can be appropriate (Note 8), and ironically, the distribution of a rank-transformed variable is grossly non-normal.

Note 8: Non-uniformity

Non-uniformity of effect or error in linear models can produce incorrect estimates and confidence limits. Check for non-uniformity by comparing standard deviations of the dependent variable in different subgroups or by examining plots of the dependent variable or its residuals for differences in scatter (heteroscedasticity) with different predicted values and/or different values of the predictors.

Differences in standard deviations or errors between groups can be taken into account for simple comparisons of means by using the unequal-variances t statistic. With more complex models use mixed modeling to allow for and estimate different standard deviations in different groups or with different treatments. For a simpler robust approach with independent subgroups, perform separate analyses then compare the outcomes using a spreadsheet (Hopkins, 2006b).

Transformation of the dependent variable is another approach to reducing non-uniformity, especially when there are differences in scatter for different predicted values. For many dependent variables, effects and errors are uniform when expressed as factors or percents; log transformation converts these to uniform additive effects, which can be modeled linearly then expressed as factors or percents after back transformation. Always use log transformation for such variables, even when a narrow range in the dependent variable effectively eliminates non-uniformity.

Rank transformation eliminates non-uniformity for most dependent variables and models, but it results in loss of precision with a small sample size and should therefore be used as a last resort. To perform the analysis, sort all observations by the value of the dependent variable, assign each observation a rank (consecutive integer), then use the rank as the dependent variable in a liner model. Such analyses are often referred to incorrectly as nonparametric.

Use the transformed variable, not the raw variable, to gauge magnitudes of correlations and of standardized differences or changes in means. Back-transform the mean effect to a mean in raw units and its confidence limits to percents or factors (for log transformation) or to raw units at the mean of the transformed variable or at an appropriate value of the raw variable (for all other transformations). When analysis of a transformed variable produces impossible values for an effect or a confidence limit (e.g., a negative rank with the rank transformation), the assumption of normality of the sampling distribution of the effect is violated and the analysis is therefore untrustworthy. Appropriate use of bootstrapping avoids this problem.

Note 9: Outliers

Outliers for a continuous dependent variable represent a kind of non-uniformity that appears on a plot of residuals vs predicteds as individual points with much larger residuals than other points. To delete the outliers in an objective fashion, set a threshold by first standardizing the residuals (dividing by their standard deviation). The resulting residuals are t statistics, and with the assumption of normality, a threshold for values that would occur rarely (<5% of the time is a good default) depends on sample size. Approximate sample sizes and thresholds for the absolute value of t are: <~50, >3.5; ~500, >4.0; ~5000, >4.5; ~50,000, >5.0. Some packages identify outliers more accurately using statistics that account for the lower frequency of large residuals further away from the mean predicted value of the dependent.

Note 10: Effect of Continuous Predictors

The use of two standard deviations (SD) to gauge the effect of a continuous predictor ensures congruence between Cohen's threshold magnitudes for correlations and standardized differences (Note 1). Two SD of a normally distributed predictor also corresponds approximately to the mean separation of lower and upper tertiles (2.2 SD). The SD is ideally the variation in the predictor after adjustment for other predictors; the effect of 2 SD in a correlational study is then equivalent to, and can be replaced by, the partial correlation (the square root of the fraction of variance explained by the predictor after adjustment for all other predictors).

A grossly skewed predictor can produce incorrect estimates or confidence limits, so it should be transformed to reduce skewness. Log transformation is often suitable for skewed predictors that have only positive values; as simple linear predictors their effects are then expressed per factor or percent change of their original units. Alternatively, a skewed predictor can be parsed into quantiles (usually 2-5 subgroups with equal numbers of observations) and included in the model as a nominal variable or as an ordinal variable (a numeric variable with integer values). Parsing is also appropriate for a predictor that is likely to have a non-linear effect not easily or realistically modeled as a polynomial.

Note 11: SEM vs SD

The standard error of the mean (SEM = SD/V(group sample size)) is the sampling variation in a group mean, which is the expected typical variation in the mean from sample to sample. Some researchers argue that, as such, this measure communicates uncertainty in the mean and is therefore preferable to the SD. A related widespread belief is that non-overlap of SEM bars on a graph indicates a difference that is statistically significant at the 5% level. Even if statistical significance was the preferred approach to inferences, this belief is justified only when the SEM in the two groups are equal, and for comparisons of changes in means, only when the SEM are for means of change scores. Standard error bars on a time-series graph of means of repeated measurements thus convey a false impression of significance or non-significance, and therefore, to avoid confusion, SEM should not be shown for any data. In any case, researchers are interested not in the uncertainty in a single mean but in the uncertainty of an effect involving means, usually a simple comparison of two means. Confidence intervals or related inferential statistics are used to report uncertainty in such effects, making the SEM redundant and inferior.

The above represents compelling arguments for not using the SEM, but there are even more compelling arguments for using the SD. First, it helps to assess non-uniformity, which manifests as different SD in different groups. Secondly, it can signpost the likely need for log transformation, when the SD of a variable that can have only positive values is of magnitude similar to or greater than the mean. Finally and most importantly, the SD communicates the magnitude of differences or changes between means, which by default should be assessed relative to the usual between-subject SD (Note 1). The manner in which the SEM depends on sample size makes it unsuitable for any of these applications, whereas the SD is practically unbiased for sample sizes ~10 or more (Gurland and Tripathi, 1971).

Note 12: Error-related Bias

Random error or random misclassification in a variable attenuates effects involving the variable and widens the confidence interval. (Exception: random error in a continuous dependent variable does not attenuate effects of predictors on means of the variable.) After adjustment of the variable for any systematic difference from a criterion in a validity study with subjects similar to those in your study, it follows from statistical first principles that the correction for attenuation of an effect derived directly from the variable's coefficient in a linear model is 1/[v.sup.2], where v is the validity correlation coefficient; the correction for a correlation with the variable is 1/v. In this context, a useful estimate for the upper bound of v is the square root of the short-term reliability correlation.

When one variable in an effect has systematic error or misclassification that is substantially correlated with the value of the other variable, the effect will be biased up or down, depending on the correlation. Example: a spurious beneficial effect of physical activity on health could arise from healthier people exaggerating their self-reported activity.

Substantial random or systematic error of measurement in a covariate used to adjust for confounding results in partial or unpredictable adjustment respectively and thereby renders untrustworthy any claim about the presence or absence of the effect after adjustment. This problem applies also to a mechanisms analysis involving such a covariate.

Note 13: Limits of Agreement

Bland and Altman introduced limits of agreement (defining a reference interval for the difference between measures) and a plot of subjects' difference vs mean scores of the measures (for checking relative bias and non-uniformity) to address what they thought were shortcomings arising from misuse of validity and reliability correlation coefficients in measurement studies. Simple linear regression nevertheless provides superior statistics in validity studies, for the following reasons: the standard error of the estimate and the validity correlation can show that a measure is suitable for clinical assessment of individuals and for sample-based research, yet the measure would not be interchangeable with a criterion according to the limits of agreement; the validity correlation provides a correction for attenuation (see Note 12), but no such correction is available with limits of agreement; the regression equation provides trustworthy estimates of the bias of one measure relative to the other, whereas the Bland-Altman plot shows artifactual bias for measures with substantially different errors (Hopkins, 2004); regression statistics can be derived in all validity studies, whereas limits of agreement can be derived from difference scores in only a minority of validity studies ("method-comparison" studies, where both measures are in the same units); finally, limits of agreement in a method-comparison study of a new measure with an existing imprecise measure provide no useful information about the validity of the new measure, whereas regression validity statistics can be combined with published validity regression statistics for the imprecise measure to correctly estimate validity regression statistics for the new measure.

Arguments have also been presented against the use of limits of agreement as a measure of reliability (Hopkins, 2000). Additionally, data generally contain several sources of random error, which are invariably estimated as variances in linear models then combined and expressed as standard errors of measurement and/or correlations. Transformation to limits of agreement is of no further clinical or theoretical value.

Note 14: Qualitative Inferences

Some qualitative researchers believe that it is possible to use qualitative methods to generalize from a sample of qualitatively analyzed cases (or assessments of an individual) to a population (or the individual generally). Others do not even recognize the legitimacy of generalizing. In our view, generalizing is a fundamental obligation that is best met quantitatively, even when the sample is a series of qualitative case studies or assessments.

Acknowledgement: Chris Bolter, Janet Dufek, Doug Curran-Everett, Patria Hume, George Kelley, Ken Quarrie, Chris Schmid, David Streiner and Martyn Standage provided valuable feedback on drafts, as did nine reviewers on the submitted manuscript. The authors have no professional relationship with a for-profit organization that would benefit from this study; publication does not constitute endorsement by ACSM. No funding was received for this work from any organization, other than salary support for the authors from their respective institutions.


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Published Nov 2009. [c] 2009

Will G Hopkins (1), Alan M. Batterham (2), Stephen W Marshall (3), Juri Hanin (4)

(1) Institute of Sport and Recreation Research, AUT University, Auckland NZ, Email; (2) School of Health and Social Care,University of Teesside, Middlesbrough UK, Email; (3) Departments of Epidemiology, Orthopedics, and Exercise & SportScience, University of North Carolina at Chapel Hill, Chapel Hill NC, Email; (4) KIHU-Research Institute for Olympic Sports,Jyvaskyla, Finland, Email. Reviewer: Ian Shrier, Department of Family Medicine, McGill University, Montreal, Canada.

An earlier version of this article was published in the January 2009 issue of Medicine and Science in Sports and Exercise. This update indicates changes highlighted in pale green. Cite the current article for reference to such changes. Cite the earlier article (Hopkins et al., 2009) for reference to unchanged material.
TABLE 1. Statements of best practice for reporting research

TABLE 2. Generic statistical advice for sample-based studies
    Subject Characteristics
    Outcome Statistics
  Note 1: Inferences
  Note 2: Access to Data
  Note 3: Multiple Inferences
  Note 4: Sample Size
  Note 5: Mechanisms
  Note 6: Linear Models
  Note 7: Non-parametric Analysis
  Note 8: Non-uniformity
  Note 9: Outliers
  Note 10: Effect of Continuous Predictors
  Note 11: SEM vs SD
  Note 12: Error-related Bias
  TABLE 3. Additional statistical advice for specific designs
      Subject Characteristics
      Outcome Statistics: Continuous Dependents
      Outcome Statistics: Event Dependents
      Outcome Statistics
      Outcome Statistics
      Study Characteristics
      Case Description
      Results and Discussion
  Note 13: Limits of Agreement
  Note 14: Qualitative Inferences

Table 1. Recent statements of best practice for reporting
various kinds of biomedical research.

Interventions (experiments)

CONSORT: Consolidated Standards of Reporting Trials
(Altman et al., 2001; Moher et al., 2001). See
for statements, explanations and extensions
to abstracts and to studies involving equivalence
or non-inferiority, clustered randomization, harmful
outcomes, non-randomized designs, and various kinds
of intervention.

Observational (non-experimental) studies

STROBE: Strengthening the Reporting of Observational
Studies in Epidemiology (Vandenbroucke et al., 2007;
von Elm et al., 2007). See for
statements and explanations, and see for
extension to gene-association studies.

Diagnostic tests

STARD: Standards for Reporting Diagnostic Accuracy
(Bossuyt et al., 2003a; Bossuyt et al., 2003b).


QUOROM: Quality of Reporting of Meta-analyses
(Moher et al., 1999). MOOSE: Meta-analysis of Observational
Studies in Epidemiology (Stroup et al., 2000).
See also the Cochrane Handbook (at and
guidelines for meta-analysis of diagnostic tests (Irwig et
al., 1994) and of gene-association studies (at

TABLE 2. Generic statistical advice for sample-based studies.


* State why you studied the effect(s).

* State the design, including any randomizing and blinding.

* Characterize the subjects who contributed to the estimate
of the effect(s) (final sample size, sex, skill,

* Ensure all numbers are either in numeric or graphical

form in the Results section of the manuscript.

* Show magnitudes and confidence intervals or limits of
the most important effect(s). Avoid P values. [Note 1]

* Make a probabilistic statement about clinical, practical,
or mechanistic importance of the effect(s).

* The conclusion must not be simply a restatement of


* Explain the need for the study.

--Justify choice of a particular population of subjects.

--Justify choice of design here, if it is one of the reasons
for doing the study.

* State an achievable aim or resolvable question about
the magnitude of the effect(s). Avoid hypotheses.
[Note 1]


* Explain the recruitment process and eligibility criteria for
acquiring the sample from a population.

--Justify any stratification aimed at proportions of subjects
with certain characteristics in the sample.

* Include permission for public access to depersonalized
raw data in your application for ethics approval. [Note 2]


* Describe any pilot study aimed at measurement properties
of the variables and feasibility of the design.

* To justify sample size, avoid adequate power for statistical
significance. Instead, estimate or reference the
smallest important values for the most important effects
and use with one or more of the following approaches,
taking into account any multiple inferences and quantification
of individual differences or responses
[Notes 3, 4]:

--adequate precision for a trivial outcome, smallest expected
outcome, or comparison with a published outcome;

--acceptably low rates of wrong clinical decisions;

--adequacy of sample size in similar published studies;

--limited availability of subjects or resources (in which
case state the smallest magnitude of effect your study
could estimate adequately).

* Detail the timings of all assessments and interventions.

* See also Table 3 for advice on design of specific kinds
of study.


* Justify choice of dependent and predictor variables in
terms of practicality and measurement properties specific
to the subjects and conditions of the study. Use
variables with the smallest errors.

* Justify choice of potential moderator variables: subject
characteristics or differences/changes in conditions or
protocols that could affect the outcome and that are included
in the analysis as predictors to reduce confounding
and account for individual differences.

* Justify choice of potential mediator variables: measures
that could be associated with the dependent variable
because of a causal link from a predictor and that are
included in an analysis of the mechanism of the effect of
the predictor. [Note 5]

* Consider including open-ended interviews or other
qualitative methods, which afford serendipity and flexibility
in data acquisition.

--Use in a pilot phase aimed at defining purpose and
methods, during data gathering in the project itself, and
in a follow-up assessment of the project with stake


* Describe any initial screening for miscodings, for example
using stem-and-leaf plots or frequency tables.

* Justify any imputation of missing values and associated
adjustment to analyses.

* Describe the model used to derive the effect. [Note 6]

--Justify inclusion or exclusion of main effects, polynomial
terms and interactions in a linear model.

--Explain the theoretical basis for use of any non-linear

--Provide citations or evidence from simulations that any
unusual or innovative data-mining technique you used
to derive effects (neural nets, genetic algorithms, decision
trees, rule induction) should give trustworthy estimates
with your data.

--Explain how you dealt with repeated measures or other
clustering of observations.

* Avoid purely non-parametric analyses. [Note 7]

* If the dependent variable is continuous, indicate whether
you dealt with non-uniformity of effects and/or error by
transforming the dependent variable, by modeling different
errors in a single analysis, and/or by performing and
combining separate analyses for independent groups.
[Note 8]

* Explain how you identified and dealt with outliers, and
give a plausible reason for their presence. [Note 9]

* Indicate how you dealt with the magnitude of the effect
of linear continuous predictors or moderators, either as
the effect of 2 SD, or as a partial correlation, or by parsing
into independent subgroups. [Note 10]

* Indicate how you performed any subsidiary mechanisms
analysis with potential mediator variables, either using
linear modeling or (for interventions) an analysis of
change scores. [Note 5]

* Describe how you performed any sensitivity analysis, in
which you investigated quantitatively, either by simulation
or by simple calculation, the effect of error of measurement
and other potential sources of bias on the
magnitude and uncertainty of the effect statistic(s).

* Explain how you made inferences about the true (infinite-sample)
value of each effect. [Note 1]

--Show confidence intervals or limits.

--Justify a value for the smallest important magnitude,
then base the inference on the disposition of the confidence
interval relative to substantial magnitudes.

--For effects with clinical or practical application, make a
decision about utility by estimating chances of benefit
and harm.

- Avoid the traditional approach of statistical significance
based on a null-hypothesis test using a P value.

--Explain any adjustment for multiple inferences.
[Note 3]

* Include this statement, when appropriate: measures of
centrality and dispersion are mean [+ or -] SD.

--Add the following statement, when appropriate: for
variables that were log transformed before modeling,
the mean shown is the back-transformed mean of the
log transform, and the dispersion is a coefficient of
variation (%) or */?*? factor SD.

--The range (minimum-maximum) is sometimes informative,
but beware that it is strongly biased by sample

--Avoid medians and other quantiles, except when parsing
into subgroups.

--Never show standard errors of means. [Note 11]

* See also Table 3 for advice on analysis of specific kinds
of study.


Subject Characteristics

* Describe the flow of number of subjects from those who
were first approached about participation through those
who ended up providing data for the effects.

* Show a table of descriptive statistics of variables in
important groups of the subjects included in the final
analysis, not the subjects you first recruited.

--For numeric variables, show mean [+ or -] SD. [Note 11]

--For nominal variables, show percent of subjects.

--Summarize the characteristics of dropouts (subjects
lost to follow-up) if they represent a substantial proportion
(>10%) of the original sample or if their loss is
likely to substantially bias the outcome. Be precise
about which groups they were in when they dropped
out and why they dropped out.

* See also Table 3 for advice on reporting subject characteristics
in specific kinds of study.

Outcome Statistics

* Avoid all exact duplication of data between tables,
figures, and text.

* When adjustment for subject characteristics and other
potential confounders is substantial, show unadjusted
and adjusted outcomes.

* Use standardized differences or changes in means to
assess qualitative magnitudes of the differences, but
there is generally no need to show the standardized
values. [Note 1]

* If the most important effect is unclear, provide a qualitative
interpretation of its uncertainty. (For example, it is
unlikely to have a small beneficial effect and very
unlikely to be moderately beneficial.) State the approximate
sample size that would be needed to make it

* See also Table 3 for advice on outcome statistics in
specific kinds of study.


* Use the following abbreviations for units: km, m, cm,
mm, pm, L, ml, pL, kg, g, mg, pg, pg, y, mo, wk, d, h, s,
ms, A, mA, [micro]A, V, mV, [micro]V, N, W, J, kJ, MJ, [degrees],
[degrees]C, rad, kHz, Hz, mol, mmol, osmol, mosmol.

* Insert a space between numbers and units, with the
exception of % and [degrees]. Examples: 70
mlx[min.sup.-1]x[kg.sup.-1]; 90%.

* Insert a hyphen between numbers and units only when
grammatically necessary: the test lasted 4 min; it was a
4-min test.

* Ensure that units shown in column or row headers of a
table are consistent with data in the cells of the table.

* Round up numbers to improve clarity.

--Round up percents, SD, and the "[+ or -]" version of confidence
limits to two significant digits. A third digit is
sometimes appropriate to convey adequate accuracy
when the first digit is "1"; for example, 12.6% vs 13%.
A single digit is often appropriate for small percents
(<1%) and some subject characteristics.

--Match the precision of the mean to the precision of the
SD. In these properly presented examples, the true
values of the means are the same, but they are
rounded differently to match their different SD: 4.567 [+ or -]
0.071, 4.57 [+ or -] 0.71, 4.6 [+ or -] 7.1, 5 [+ or -] 71, 0
[+ or -] 710, 0 [+ or -] 7100.

--Similarly, match the precision of an effect statistic to
that of its confidence limits.

* Express a confidence interval using "to" (e.g., the effect
was 3.2 units; 90% confidence interval -0.3 to 6.7 units)
or express confidence limits using "[+ or -]" (3.2 units; 90%
confidence limits [+ or -]3.5 units).

--Drop the wording "90% confidence interval/limits" for
subsequent effects, but retain consistent punctuation
(e.g., 2.1%; [+ or -]3.6%). Note that there is a semicolon or
comma before the "[+ or -]" and no space after it for confidence
limits, but there is a space and no other punctuation
each side of a "[+ or -]" denoting an SD. Check your
abstract and results sections carefully for consistency
of such punctuation.

--Confidence limits for effects derived from back-transformed
logs can be expressed as an exact
xMactor by taking the square root of the upper limit
divided by the lower limit. Confidence limits of measurement
errors and of other SD can be expressed in
the same way, but the resulting */-factor becomes
less accurate as degrees of freedom fall below 10.

* When effects and confidence limits derived via log
transformation are less than ~[+ or -]25%, show as percent
effects; otherwise show as factor effects. Examples: 3%,
-14 to 6%; 17%, [+ or -]6%; a factor of 0.46, 0.18 to
1.15; a factor of 2.3, x/-1.5.

* Do not use P-value inequalities, which oversimplify
inferences and complicate or ruin subsequent meta-analysis.

--Where brevity is required, replace with the [+ or -] or x/+
form of confidence limits. Example: "active group 4.6
units, control group 3.6 units (P>0.05)" becomes "active
group 4.6 units, control group 3.6 units (95% confidence
limits [+ or -]1.3 units)".

--If you accede to an editor's demand for P values, use
two significant digits for P>0.10 and one for P<0.10.
Examples: P=0.56, P=0.10, P=0.07, P=0.003,
P=0.00006 (or 6E-5).


* Use figures sparingly and only to highlight key outcomes.

* Show a scattergram of individual values or residuals
only to highlight the presence and nature of unusual
non-linearity or non-uniformity.

--Most non-uniformity can be summarized non-graphically,
succinctly and more informatively with appropriate
SD for appropriate subgroups.

--Do not show a scattergram of individual values that
can be summarized by a correlation coefficient. (Exception:
validity studies.)

* Use line diagrams for means of repeated measurements.
Use bar graphs for single observations of means
of groups of different subjects.

* In line diagrams and scattergrams, choose symbols to
highlight similarities and differences in groups or treatments.

--Make the symbols too large rather than too small.

--Explain the meaning of symbols using a key on the
figure rather than in the legend.

--Place the key sensibly to avoid wasting space.

--Where possible, label lines directly rather than via a

* Use a log scale for variables that required log transformation
when the range of values plotted is greater than

* Show SD of group means to convey a sense of magnitude
of effects. [Note 11]

--For mean change scores, convey magnitude by showing
a bar to the side indicating one SD of composite
baseline scores.

* In figures summarizing effects, show bars for confidence
intervals rather than asterisks for P values.

--State the level of confidence on the figure or in the

--Where possible, show the range of trivial effects on the
figure using shading or dotted lines. Regions defining
small, moderate and large effects can sometimes be
shown successfully.


* Avoid restating any numeric values exactly, other than
to compare your findings with those in the literature.

* Avoid introducing new data.

* Be clear about the population to which your effect statistics
apply, but consider their wider applicability.

* Interpret a mechanisms analysis cautiously. [Note 5]

* Assess the possible bias arising from the following

--confounding by non-representativeness or imbalance
in the sampling or assignment of subjects, when the
relevant subject characteristics could affect the dependent
variable and have not been adjusted for by inclusion
in the model;

--random or systematic error in a continuous variable or
classification error in a nominal variable; [Note 12]

--choosing the largest or smallest of several effects that
have overlapping confidence intervals; [Note 3]

--your prejudices or desire for an outcome, which can
lead you to filter data inappropriately and misinterpret

TABLE 3. Additional statistical advice for specific sample-based
and single-case designs.


* Justify any choice between pre-post vs post-only and
between parallel-group vs crossover designs. Avoid
single-group (uncontrolled) designs if possible. See
Reference (Batterham and Hopkins, 2005) for more.

* Investigate more than one experimental treatment only
when sample size is adequate for multiple comparisons.
[Note 4]

* Explain any randomization of subjects to treatment
groups or treatment sequences, any stratification (balancing
of numbers in subject-characteristic subgroups),

and any minimization of differences of means of subject
characteristics in treatment groups. State whether/how
randomization to groups or sequences was concealed
from researchers.

* Detail any blinding of subjects and researchers.

* Detail the timing and nature of assessments and interventions.


* Indicate how you included, excluded or adjusted for
subjects who showed substantial non-compliance with
protocols or treatments or who were lost to follow-up.

* In a parallel-groups trial, estimate and adjust for the

potential confounding effect of any substantial differences
in mean characteristics between groups.
--In pre-post trials in particular, estimate and adjust for
the effect of baseline score of the dependent variable
on the treatment effect. Such adjustment eliminates
any effect of regression to the mean, whereby a difference
between groups at baseline arising from error of
measurement produces an artifactual treatment effect.

Subject Characteristics

* For continuous dependent and mediator variables, show
mean and SD in the subject-characteristics table only at

Outcome Statistics: Continuous Dependents

(For event-type dependents, see the section below on
prospective cohort studies.)

* Baseline means and SD that appear in text or a table
can be duplicated in a line diagram summarizing means
and SD at all assay times

* Show means and SD of change scores in each group.

* Show the unadjusted and any relevant adjusted differences
between the mean changes in treatment and control
(or exposed and unexposed) groups, with confidence

* Show the standard error of measurement derived from
repeated baseline tests and/or pre-post change scores
in a control group.

* Include an analysis for individual responses derived
from the SD of the change scores. In post-only crossovers
this analysis requires an assumption about, or
separate estimation of, error of measurement over the
time between treatments.


* If there was lack or failure of blinding, estimate bias due
to placebo and nocebo effects (outcomes better and
worse than no treatment arising purely from expectation
with the experimental and control treatments respectively).



* Describe the methods of follow-up.

* Indicate how you included, excluded or adjusted for
subjects who showed substantial non-compliance with
protocols or treatments or who were lost to follow-up.

* Adjust effects for any substantial difference between
groups at baseline.

Outcome Statistics: Event Dependents

(For continuous dependents, see the section above on

* When the outcome is assessed at fixed time points,
show percentage of subjects in each group who experienced
the event at each point.

* When subjects experience multiple events, show raw or
factor means and SD of counts per subject.

* When the outcome is time to event, display survival
curves for the treatment or exposure groups.

* Show effects as the risk, odds or hazard ratios adjusted
for relevant subject characteristics.

--Present them also in a clinically meaningful way by
making any appropriate assumptions about incidence,
prevalence, or exposure to convert the ratios to risks
(proportions affected) and risk difference between
groups or for different values of predictors, along with
confidence limits for the risk ratio and/or risk difference
(Hopkins et al., 2007).

--Adjusted mean time to event and its ratio or difference
between groups is a clinically useful way to present
some outcomes.


* Take into account the fact that confounding can bias the
risk ratio by as much as [??]-2.0-3.0 in most cohort and
case-control studies (Taubes, 1995).


* Explain how you tried to choose controls from the same
population giving rise to the cases.

* Justify the case:control ratio. (Note that >5 controls per
case or >5 cases per control give no useful increase in

* Case-crossovers: describe how you defined the time
windows for assessing case and control periods.

Outcome Statistics

* Present risk-factor outcomes in a clinically meaningful
way by converting the odds ratio (which is a hazard ratio
with incidence density sampling) to a risk ratio and/or
risk difference between control and exposed subjects in
an equivalent cohort study over a realistic period
(Hopkins et al., 2007).


* See the Discussion point on confounding in cohort

* Estimate bias due to under-matching, over-matching or
other mis-matching of controls.

Outcome Statistics

* Show simple unadjusted effects and effects adjusted for
all other predictors in the model.



* Specify the measurement and structural models using a
path diagram.

* Explain the estimation method and the strategy for
assessing goodness of fit.

* Demonstrate that all parameters were estimable.


* Justify the cost-effectiveness of the criterion measure,
citing studies of its superiority and measurement error.


* Use linear or non-linear regression to estimate a calibration
equation, a standard error of the estimate, and a
validity correlation coefficient.

--For criterion and practical measures in the same metric,
use the calibration equation to estimate bias in the
practical measure over its range.

--Do not calculate limits of agreement or present a
Bland-Altman plot. [Note 13]


* Document the diagnostic accuracy of the method or
combination of methods used as the reference standard.


* Calculate the following diagnostic measures, all of which
can be useful: the validity correlation coefficient, the
kappa coefficient, sensitivity, specificity, positive and
negative predictive values, positive and negative diagnostic
likelihood ratios, and diagnostic odds ratio.

* For a continuous measure, calculate area under the
ROC curve and give the above diagnostic measures for
an appropriate threshold.


* Justify number of trials, raters, items of equipment
and/or subjects needed to estimate the various within
and between standard deviations.

* Justify times between trials to establish effects due to
familiarization (habituation), practice, learning, potentialtion,
and/or fatigue.


* Assess habituation and other order-dependent effects in
simple reliability studies by deriving statistics for consecutive
pairs of measurements.

* The reliability statistics are the change in the mean
between measurements, the standard error of measurement
(typical error), and the appropriate intraclass
correlation coefficient (or the practically equivalent test-retest
Pearson correlation).

--Do not abbreviate standard error of measurement as
SEM, which is confused with standard error of the

--Avoid limits of agreement. [Note 13]

* With several levels of repeated measurement (e.g.,
repeated sets, different raters for the same subjects)
use judicious averaging or preferably mixed modeling to
estimate different errors as random effects.


* Describe any pilot work with experts and subjects to
develop or modify wording in any exploratory factor


* Specify the analytic approach (principal components or
principal axes), the criteria used to extract factors, the
rotation method and factor-loading cutoffs for selection
of variables for each factor, and the communalities to
justify exclusion of items from the instrument.

* For confirmatory factor analysis use an appropriate
structural equation model.

* For each factor calculate the square root of Cronbach's
alpha, which is an upper bound for the validity correlation.


* Describe the search strategy and inclusion criteria for
identifying relevant studies.

* Explain why you excluded specific studies that other
researchers might consider worthy of inclusion.


* Explain how you reduced study-estimates to a common

--Conversion to factor effects (followed by log transformation)
is often appropriate for means of continuous

--Avoid standardization (dividing each estimate by the
between-subject SD) until after the analysis, using an
appropriate between-subject composite SD derived
from some or all studies.

--Hazard ratios are often best for event outcomes.

* Explain derivation of the weighting factor (inverse of the
sampling variance, or adjusted sample size if sufficient
authors do not provide sufficient inferential information).

* Avoid fixed-effect meta-analysis. State how you performed
a random-effect analysis to allow for real differences
between study-estimates. With sufficient studies,
adjust for study characteristics by including them as
fixed effects, and account for any clustering of study-estimates
by including extra random effects.

* Use a plot of standard error or 1/V(sample size) vs
study-estimate or preferably the t statistic of the solution
of each random effect to explore the possibility of publication
bias and to identify outlier study-estimates.

* To gauge the effect of 2 SD of predictors [Note 10]
representing mean subject characteristics, use an appropriate
mean of the between-subject SD from selected
or all studies, not the SD of the study means.

Study Characteristics

* Show a table of study characteristics, study-estimates,
inferential information (provided by authors) and confidence
limits (computed by you, when necessary).

--If the table is too large for publication, make it available
at a website or on request.

--A one-dimensional plot of effects and confidence intervals
("forest plot") represents unnecessary duplication
of data in the above table.

* Show a scatterplot of study-estimates with confidence
limits to emphasize a relationship with a study characteristic.


* Regard these as sample-based studies aimed at an
inference about the value of an effect statistic in the
population of repeated observations on a single subject.

* Justify the choice of design by identifying the closest
sample-based design.

* Take into account within-subject error when estimating
"sample size" (number of repeated observations).

* State the smallest important effect, which should be the
same as for a usual sample-based study.


* Account for trends in consecutive observations with
appropriate predictors.

--Check for any remaining autocorrelation, which will
appear as a trend in the scatter of a plot of residuals vs
time or measurement number.

--Use an advanced modeling procedure that allows for
autocorrelation only if there is a trend that modeling
can't remove.

* Make it clear that the inferences apply only to your
subject and possibly only to a certain time of year or

* Perform separate single-subject analyses when there is
more than one case. With an adequate sample of
cases, use the usual sample-based repeated-measures

Case Description

* For a difficult differential diagnosis, justify the use of
appropriate tests by reporting their predictive power,
preferably as positive and negative diagnostic likelihood


* Where possible, use a quantitative Bayesian (sequential
probabilistic) approach to estimate the likelihoods of
contending diagnoses.


* State and justify your ideological paradigm (e.g.,
grounded theory).

* Describe your methods for gathering the information,
including any attempt to demonstrate congruence of
data and concepts by triangulation (use of different

* Describe your formal approach to organizing the information
(e.g., dimensions of form, content or quality,
magnitude or intensity, context, and time (Hanin, 2003)).

* Describe how you reached saturation, when ongoing
data collection and analysis generated no new categories
or concepts.

* Describe how you solicited feedback from respondents,
peers and experts to address trustworthiness of the outcome.

* Analyze a sufficiently large sample of cases or assessments
of an individual by coding the characteristics and
outcomes of each case (assessment) into variables and
by following the advice for the appropriate sample-based
study. [Note 14]

Results and Discussion

* Address the likelihood of alternative interpretations or

* To generalize beyond a single case or assessment,
consider how differences in subject or case characteristics
could have affected the outcome.
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