Expected Frequency

An expected value equals the average or mean of some statistic, whereas an expected frequency is a mean count or mean number of times an event occurs. Events could be a category of a single discrete variable, a cell of a cross-classification of two or more discrete [Page 348]variables, or some other specified event (e.g., the number of goals scored by a team during a soccer game). An expected frequency is computed by multiplying the probability that an event occurs by the total number of possible times that the event could occur. For example, consider random samples of size n = 75 people from a population in which the probability that an individual is left-handed equals π = 0.10. The number of left-handers in a sample could be any integer between 0 and 75; however, the expected or mean number of left-handers equals nπ = 75 (0.10) = 7.5. Although frequencies are always integers, expected frequencies typically are not integers.

Expected frequencies are used in test statistics for assessing hypotheses and model goodness of fit. Two common test statistics are Pearson’s CHI-SQUARE TEST, χ2 = ΣNi=1(fi − Fi)2/Fi, and the LIKELIHOOD RATIO STATISTIC, L2 = ΣNi=1filog(fi/Fi), where fi equals the observed number of observations for category i, Fi equals the expected frequency for category i, and N equals the total number of categories. If a null hypothesis or model for data specifies a value for the probability (or probabilities), such as “the probability of being left-handed is 0.10,” then an expected frequency is computedas Fi = nπi, where n equalsthetotalnumber of observations.

The probabilities of events usually are unknown and are estimated from data by computing sample proportions or by fitting a model to data. When data are used to estimate probabilities, the resulting expected frequencies are known as estimated expected frequencies. For example, consider the data in Table 1, which is reformatted from data presented by Jon Cohen (2001). The data consist of the number of smallpox cases in Liverpool, England, during 1902–1903, cross-classified according to whether a person was vaccinated in infancy and whether the person died of smallpox. Assuming that vaccinations do not affect the severity of a case of smallpox, the probability of death as a result of smallpox is estimated by computing the proportion of smallpox cases that resulted in death, that is, the total number of deaths divided by the total number of cases, which equals ^πdeath =79/1,163 = 0.0679. The estimated expected frequency of death given vaccination equals ^Fvac, death = nvac ^πdeath = 943 (0.0679) = 64.0559. The estimated expected frequencies for other cells in Table 1 could also be computed. For a hypothesis test, the estimated expected frequencies, ^Fi, replace the expected frequencies, Fi, in Pearson’s chi-square statistic or the likelihood ratio statistic.

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