# ESCI 2001 to 2010

[November 2011]

ESCI (pronounced ‘esky’) is a set of interactive simulations that run under Microsoft Excel.

With ESCI you can

• explore many Confidence Interval (CI) concepts
• calculate and display CIs for your own data, for some simple designs
• calculate CIs for Cohen’s standardised effect size d
• explore noncentral t distributions and their role in statistical power
• use CIs for simple meta-analysis, using original or standardised units
explore all these concepts via vivid interactive graphical simulations.

The following ESCI modules go with published articles. These ESCI modules are free downloads, for non-commercial use, and were developed in Microsoft Excel 2003 or XP. (They may also run in earlier versions of Excel.) Sorry, no Mac versions yet.

ESCI-JPP is a module that allows you to calculate and display CIs for a wide variety of measures and designs. It goes with this article:

Finch, S., & Cumming, G. (2009). Putting research in context: Understanding confidence intervals from one or more studies. Journal of Pediatric Psychology. doi: 10.1093/jpepsy/jsn118

ESCI-PPS-p-intervals is a module that allows you to explore three of the figures in:

Cumming, G. (2008). Replication and p intervals: p values predict the future only vaguely, but confidence intervals do much better. Perspectives on Psychological Science, 3, 286-300.

NOTE: Excel 2007 The above module runs in Excel 2007, as well as Excel 2003. The modules below probably will not run properly in Excel 2007. (Excel 2007 runs much more slowly that Excel 2003, has many display bugs, and does not support smooth interactivity. If you still have Excel 2003, preserve it carefully!)

ESCI-IBI-levels_confid is a module that allows you to explore figures in this Teaching Statistics article, and to adjust levels of confidence:

Cumming, G. (2007). Inference by eye: Pictures of confidence intervals and thinking about levels of confidence. Teaching Statistics, 29, 89:93.

ESCI-CI-next-PMis a module that allows you to explore confidence intervals and replication. If an experiment is replicated, what is the average probability that an initial confidence interval will capture a replication mean? How does this vary for various initial confidence intervals? It accompanies:

Cumming, G., & Maillardet, R. (2006). Confidence intervals and replication: Where will the next mean fall? Psychological Methods, 11, 217-227.

ESCI-Inference-by-eye is a module that allows you to explore the figures in our ‘Inference by eye’ article, and to adjust the confidence level C for a single CI. It accompanies:

Cumming, G., & Finch, S. (2005) Inference by eye: Confidence intervals and how to read pictures of data. American Psychologist, 60, 170-180.

ESCI-APR-Simulationand ESCI-APR-Figures are modules that allow you to run a simulation to explore the ideas of PR, the probability of replication, and APR, the average probability of replication. They accompany:

Cumming, G. (2005). Understanding the average probability of replication. Comment on Killeen (2005). Psychological Science, 16, 1002-1004.

ESCI-JSMS is a module that allows you to calculate and display CIs for a two independent group design. It accompanies:

Wolfe, R., & Cumming, G. (2004) Communicating the uncertainty in research findings: Confidence intervals. Journal of Science and Medicine in Sport, 7, 138-143.

ESCI-Ustanding-Stats is a module that allows you to simulate replication of an experiment and see where replication means fall in relation to an initial CI. It also gives information about mean capture percentages of replication means, for a CI of a chosen level of confidence. It accompanies:

Cumming, G., Williams, J., & Fidler, F. (2004) Replication, and researchers’ understanding of confidence intervals and standard error bars. Understanding Statistics, 3, 299-311.Classic ESCI
The original ESCI is the ESCI-delta set of modules, developed in Microsoft Excel 97.

ESCIdeltaINDwas developed in Microsoft Excel 97, and optimised for 800 x 600 screen resolution.

Any ESCI-delta software download is a 1.2MB file. Save it to disk. Double-click the .exe file and it unzips automatically into a folder containing ESCI-delta. If there is a README file, note this first.

The rationale for ESCI-delta

Statistics reform requires wider use of confidence intervals (CIs) and effect size measures, in original and standardised units. We therefore need CIs for standardised effect sizes. Unfortunately in many cases these are not easily calculated, and require use of noncentral distributions. ESCI-delta enables you to find the CI for Cohen’s d for your own data, which requires use of noncentral t. ESCI-delta also allows you to explore noncentral t itself, and its most familiar application, the calculation of statistical power.

Statistics reform also encourages meta-analysis. The best way to explore (and teach) simple meta-analysis may be via CIs, graphically presented. ESCI-delta supports the development of ‘meta-analytic thinking’ in original measurement units and in standardised units using Cohen’s d. (You may know that d is simply a mean, or mean difference, divided by a standard deviation. So d is in units of standard deviations, like a z score.)

The Confidence Interval primer

The following CI primer article explains confidence intervals for Cohen’s d. It gives an introduction to noncentral t distributions and discusses related concepts including power and simple meta-analysis. It is illustrated with part-images from ESCI-delta.

Cumming, G., & Finch, S. (2001). A primer on the understanding, use and calculation of confidence intervals based on central and noncentral distributions. Educational and Psychological Measurement, 6, 530-572.

ESCI-delta is the set of six simulation workbooks mentioned in the Cumming & Finch article. (Further ESCI simulations are planned. Bookmark this site.)

## The simulations that make up ESCI-delta

### NonCentral t

#### Explore NonCentral t distributions and calculate accurate probabilities

The familiar t distribution has the single parameter df. NonCentral t distributions have in addition a noncentrality parameter. Here you can use the controls to see how the shape of the noncentral distribution (red) changes with df and the noncentrality parameter. Click boxes to display or hide a central distribution (blue), for comparison, and various other features. You can also display accurate tail probability values for your chosen df, noncentrality parameter, and t values. Noncentral t distributions are essential for statistical power and for CIs for Cohen’s d. For more, see our journal article A primer on the understanding, use and calculation of confidence intervals based on central and noncentral distributions.

### Power

#### Explore statistical power, which in most cases requires NonCentral t calculations

Click to display the central t distribution (blue) that applies when Ho is true, and/or the noncentral t distribution (red) that is appropriate when Ha is true. Display the two-tail rejection region for the statistical significance test, then the tail area(s) under the red curve that give statistical power. Vary df, alpha, and the noncentrality parameter (which is determined by the effect size, i.e. the distance between the null and alternative hypothesised mean values). You can also display the percentages of experiments that, if Ha is true, would be expected to give various significance test outcomes: These percentages may be surprising.

### CIjumping

#### Repeated sampling, to illustrate basic concepts of confidence intervals (CIs)

Take independent samples from a normal population. Set the population parameters and select the sample size n. Click to display or hide the population distribution and various other features. Click to take a single sample, shown as a dotplot. Means of successive samples are shown cascading down the screen. Run the simulation and see the means ‘dance’, illustrating the (surprisingly large?) extent of sampling variability for your chosen population and n. Set a confidence level and show a confidence interval for µ for each sample. Set a µo comparison value (usually = µ) for the mean and see which CIs capture this value. One key idea is that the CIs vary, not µ.

### CIoriginal

#### Calculate and display CIs for your data, for three simple experimental designs

These three simple tools allow you to enter your own data for three designs: Case 1, Single Group (pictured); Case 2, Two Independent Groups; and Case 3, Paired Data. The data are shown as a dotplot. For Case 1 you select a µo value as a reference value for the mean. The CI, based on your dataset, for the population mean is shown. Noting whether the CI captures µo is equivalent to conducting the corresponding t test. These tools use original measurement units throughout and are provided as an introductory step towards the more complex CIdelta, in which standardised units (Cohen’s d) are shown alongside original units.

### CIdelta

#### Calculate and display CIs for standardised effect sizes (Cohen’s d), for One and Two Group designs. Requires use of NonCentral t distributions

Enter your data for Case 1, Single Group (pictured), or Case 2, Two Independent Groups. The dotplot and CI are shown, in original units, as in CIoriginal. Set a µo comparison value, which serves as the zero for measurement of Cohen’s d, the standardised effect size. Click to display the CI for d, but note that this is not yet accurate. To find accurate values, the red noncentral t distributions must be positioned so that their tails that overlap d are of size [alpha/2]. To achieve this you can move the sliders, or you can click the buttons to have Excel do the work. For more, see our journal article A primer on the understanding, use and calculation of confidence intervals based on central and noncentral distributions.

### MAthinking

#### Explore simple Meta-analysis, based on graphical display of CIs, for effect sizes in original measurement units, and for standardised effect sizes (Cohen’s d)

The most basic meta-analysis consists of simple pooling over studies, weighted by sample size. This may be carried out in original units, or in standardised units. A simulation is provided for each of these, for the Case 1, Single Group design. For standardised units (pictured), enter the d and n values for up to 10 previous studies. Click the buttons to have Excel find and display the CI for d for each study, and for those studies combined. You can also enter d and n for an additional study (perhaps our current study) and click to show the CI for this study, and for all studies combined. Click to show the statistical significance test results: Exploration of examples will probably suggest that effect sizes (and sample sizes) are much more influential and informative than significance levels. For more, see our journal article A primer on the understanding, use and calculation of confidence intervals based on central and noncentral distributions.

*ESCI and the CI primer article use d for the sample statistic of Cohen’s effect size, and [lowercase Greek delta] for the poulation parameter. On this site we cannot display the symbol delta reliably, so we use d instead.

### Acknowledgements

Michael Smithson raised noncentral distributions as a topic of interest.
Bruce Thompson prompted the Cumming & Finch paper, and advised at every stage.
Rodney Carr showed what Excel can do.
Fiona Fidler, Sue Finch and Neil Thomason collaborated and advised.
Joanna Leeman advised, and suggested the name ESCI.
Geoff Robinson provided the routines for calculating noncentral t.
David Walsh developed the graphics, built this site and advised.