# Differential Item Functioning DIF Sample Size Nomogram

Differential Item Functioning (DIF) compares the difficulty of an item for a person sample of interest, the Focal Group (F), with the difficulty of the item for another group, the Reference Group (R). As we plan to investigate DIF, several questions arise:

• 1) What difference between the two item difficulties, i.e., what DIF size, is large enough to have substantive consequences? If we lack better information about the situation, we may choose the DIF size specified by Educational Testing Service which is 1 Delta δ unit = 0.426 logits (see RMT 20:3 p. 1070), rounded up to 0.5 logits for sample-size purposes.

• 2) What difference between item difficulties is unlikely to be merely a chance result, i.e., when is the DIF size statistically significant? We usually choose the conventional statistical significance level, p ≤ .05 (Fisher, 1926).

• 3) How many members of the Focal and Reference Groups are required for the DIF size specified in (1) to meet the DIF significance criterion specified in (2)? What samples sizes are required? This nomogram provides a guide.

The nomogram is based on a Student's t-statistic with two independent groups of different sizes, NF and NR, but equal standard deviations, S. The t-statistic of the DIF size, D, is

Group size values are chosen such that the two-sided probability of observing t or greater is p = .05 for the specified DIF size, D. The standard deviation, S, is chosen at a reasonable value, 1.88 logits, based on inspection of empirical data.

Example: if our Focal group size is 100, and our Reference group size is 1000, then the DIF size will need to be 0.4 logits or more to be statistically significant.

John Michael Linacre

Fisher R.A. (1926). The arrangement of field experiments. Journal of the Ministry of Agriculture, 33: 504.

Tristan, A. (2006). An adjustment for sample size in DIF analysis. Rasch Measurement Transactions, 20:3, p. 1070-71.

Revised 11 November 2015, with acknowledgement to Siddhi Pittayachawan.

Linacre J.M. (2013) Differential Item Functioning DIF Sample Size Nomogram, Rasch Measurement Transactions, 26:4 p. 1391

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