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Suggestions for Presenting the Results of Data Analyses

General Considerations Concerning the Presentation of Results

The standard deviation (SD) is a descriptive statistic, and the standard error (SE) is an inferential statistic. Accordingly, the SD can be used to portray the variation observed in a sample: micro-hat = 100, SD = 25 suggests a much more variable population than does micro-hat = 100, SD = 5. The expected value (i.e., an average over a large number of replicate samples) of the SD² equals σ² and depends very little on sample size (n). The SE is useful to assess the precision (repeatability) of an estimator. For example, in a comparison of males (m) and females (f), micro-hatm = 100, SE = 2 and micro-hatf = 120, SE = 1.5 would allow an inference that the population mean value micro-hat is greater among females than among males. Such an inference rests on some assumptions, such as random sampling of a defined population. Unlike the SD, the SE decreases with increasing sample size.

When presenting results such as a b, always indicate if b is a SD or a SE or is t × SE (indicating a confidence limit), where t is from the t distribution (e.g., 1.96 if the degrees of freedom are large). If a confidence interval is to be used, give the lower and upper limits as these are often asymmetric about the estimate. Authors should be clear concerning the distinction between precision (measured by variances, standard errors, coefficients of variation, and confidence intervals) and bias (an average tendency to estimate values either smaller or larger than the parameter; see White et al. 1982:22-23).

The Methods section should indicate the (1 - α)% confidence level used (e.g., 90, 95, or 99%). Information in tables should be arranged so that numbers to be compared are close to each other. Excellent advice on the visual display of quantitative information is given in Tufte (1983). Provide references for any statistical software and specific options used (e.g., equal or unequal variances in t-tests, procedure TTEST in SAS, or a particular Bayesian procedure in BUGS). The Methods section should always provide sufficient detail so that the reader can understand what was done.

In regression, discriminant function analysis, and similar procedures, one should avoid the term independent variables because the variables are rarely independent among themselves or with the response variable. Better terms include explanatory or predictor variables (see McCullagh and Neider 1989:8).

Avoid confusing low frequencies with small sample sizes. If one finds only 4 birds on 230 plots, the proportion of plots with birds can be precisely estimated. Alternatively, if the birds are the object of study, the 230 plots are irrelevant, and the sample size (4) is very small.

It is important to separate analysis of results based on questions and hypotheses formed before examining the data from results found after sequentially examining the results of data analyses. The first approach tends to be more confirmatory, while the second approach tends to be more exploratory. In particular, if the data analysis suggests a particular pattern leading to an interesting hypothesis then, at this midway point, few statistical tests or measures of precision remain valid (Lindsey 1999 a,b; White 2000). That is, an inference concerning patterns or hypotheses as being an actual feature of the population or process of interest are not well supported (e.g., likely to be spurious). Conclusions reached after repeated examination of the results of prior analyses, while interesting, cannot be taken with the same degree of confidence as those from the more confirmatory analysis. However, these post hoc results often represent intriguing hypotheses to be readdressed with a new, independent set of data. Thus, as part of the Introduction, authors should note the degree to which the study was exploratory versus confrimatory. Provide information concerning any post hoc analyses in the Discussion section.

Statistical approaches are increasingly important in many areas of applied science. The field of statistics is a science, with new discoveries leading to changing paradigms. New methods sometimes require new ways of effectively reporting results. We should be able to evolve as progress is made and changes are necessary. We encourage wildlife researchers and managers to capitalize on modern methods and to suggest how the results from such methods might be best presented. We hope our suggestions will be viewed as constructive.

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