Bootstrapping Inferential Statistics with a Spreadsheet Will G Hopkins Sportscience 16, 12-15, 2012 (sportsci.org/2012/wghboot.htm) Sport and Recreation, AUT University, Auckland 0627, New
Zealand. Email.
Reviewer: Alan M Batterham, School of Health and Social Care, University of
Teesside, Middlesbrough TS1 3BA, UK. |

When you analyze the data from the study
of a sample, you have to estimate the uncertainty in the magnitude of an
effect representing the relationship between predictor and dependent variables.
Fine, so how do you calculate the
confidence limits and the probabilities? The usual approach is to make
assumptions about the way the value of the effect statistics would vary, if
you repeated the study again and again. The values make up the so-called Bootstrapping is an alternative approach
to generating confidence limits and probabilities about the true value of the
effect, and it is the
The term For a sample of 20 or more subjects drawn
randomly from some population, you can "sort of" recreate the
population by duplicating the sample endlessly. The next step is to draw at
least 1000 samples from this population, each of the same size as the
original sample. In any given sample, some subjects will appear twice or
more, while others won't be there at all, but that doesn't matter. Next you
calculate the values of the outcome statistics for each of these samples. In
the example above, the statistics would be the value of dose at the maximum,
given by x = ‑ The median value (50th percentile) from the bootstrap samples should be practically the same as the value of the outcome statistic in the original sample. A slight mismatch can occur with only 1000 bootstrap samples. I have not used more samples, because the files are already quite large (2-3 MB), and the calculations can be slow to update. When you refresh the bootstrap samples (using Ctrl-D–see instructions in the spreadsheet), the median should hover around the original value. A consistent substantial difference can arise when the dependent and/or predictor variables are skewed, in which case log transformation may correct the problem. A predictor variable with only a few integer values (e.g., 0 and 1, denoting females and males) can also result in a consistent mismatch, especially if most of the values of the predictor are the same (e.g., 17 males denoted by 1s and only 3 females denoted by 0s). If something like this in your data produces a big difference, the only solution is a larger sample size than in your original study, which is usually out of the question by the time you are doing the analysis. The links below point to four spreadsheets. I suggest you work your way through them in the order shown. Start with the simplest of all linear models, a single predictor. Try changing the values of the predictor to 0s and 1s to model the simple difference in the means between two groups. (Note that bootstrapping automatically takes into account any difference in the standard deviations in the two groups, which you would normally deal with using the unequal-variances t statistic.) Move on to the spreadsheet for a quadratic predictor. You will find this spreadsheet allows for a quadratic maximum (an inverted-U shape) and a quadratic minimum (a U shape). If the quadratic effect is weak and the sample size is not large, a substantial proportion of the bootstrap samples will have a shape opposite to that of the original sample, in which case you will have to abandon quadratic modeling and opt instead for a simple linear model (the previous spreadsheet or usual modeling). The third spreadsheet has a quadratic model with adjustment for the effect of an extra linear covariate. The covariate can be continuous or scored simply as 0s and 1s, as shown in the spreadsheet. Note, however, that the model fits a quadratic of the same shape to the two groups of subjects implied by the 0s and 1s. If you want to fit a different quadratic to two or more groups, put each group into a separate spreadsheet, hopefully with 20 subjects in each group! You can do inferential comparisons of the resulting statistics for the groups using the spreadsheet to compare/combine effects (Hopkins, 2006). The last spreadsheet has two linear predictors. Use this one if quadratic modeling with the third spreadsheet fails. For all their complexity, these spreadsheets lack several features of my other spreadsheets: log transformation, standardization of effects, and qualitative inferences… • You will need to do any necessary log transformation before entering the numbers in the bootstrap spreadsheet. If your data represent a change in performance (as shown in the spreadsheets) and the effects are more than a few percent, you should enter the change in 100× the natural log of the performance scores, not the actual percent changes. Back-transform the bootstrapped effect on performance to a percent score using the formula 100*exp(effect/100)-100. (For effects of <10%, there is practically no difference between the 100×natural-log and the back-transformed effects.) • Standardization is performed by dividing all effects by the appropriate between-subject standard deviation. If you used log transformation, do the standardization entirely with log-transformed values, including the standard deviation of the log-transformed raw data. •
The spreadsheets
provide estimates of chances that the true effects are substantial and
the odds ratios for substantially positive/negative, but you will have to
understand the process of magnitude-based inference to convert these to
qualitative inferences ( Finally, how the spreadsheets work… I use the LINEST function to do a multiple linear regression connecting the predictors to the dependent. LINEST has several annoying "features": you have to invoke it with a strange combination of keystrokes, you can't have missing values, you can't insert columns, and (unbelievably) the coefficients of the predictors are produced in the opposite order to the variables. LINEST also produces standard errors but not covariances for the coefficients, so you can't use it to estimate confidence limits for predicted values. (Bootstrapping generates confidence limits without using the standard errors.) See the link below for separate instructions on how to use LINEST. Bootstrapping is a lot easier with Excel since the advent of xlsx files, because each bootstrap sample occupies several columns, and xls files were limited to 256 columns. For each bootstrap sample I create a set of columns that are copies of the columns where the original data are analyzed. The data for each bootstrap sample are selected from the original data using the RANDBETWEEN and INDEX functions, as you will see if you click on the appropriate cells. It's then a simple matter to generate the confidence limits and probabilities of exceeding magnitude thresholds using the PERCENTILE function. ## Spreadsheets
A single linear predictor: Bootstrap1predictor.xlsx A quadratic predictor: BootstrapQuadratic.xlsx A quadratic plus a linear covariate: BootstrapQuadraticPlus1covariate.xlsx Two linear predictors: Bootstrap2predictors.xlsx How to use LINEST: UsingLINESTinExcel.xlsx ## References
Published July 2012 |