A Significantly Extreme Deviate in Data with a Non-Significant Heterogeneity Chi Square

Abstract

It has been emphasized by Yates [1948] that the usual x2 calculated to test heterogeneity in an r X s contingency table, covers all forms of departure from proportionality and so may be insensitive in detecting departures of any particular specified type. The important general problem of subdividing such heterogeneity x2's into components of interest has been described in detail for a number of cases by Fisher [1936-54] and its general mathematical justification considered by Irwin [1949] and Lancaster [1949]. The possibility of a non-significant heterogeneity x2, containing a significant component for a particular type of departure from proportionality does not seem to be as widely recognized as in the parallel situation in the analysis of variance. A situation of this type arose in the analysis of data on the proportion of pin plants (genotype ss) among segregating progenies of open pollinated homostyle plants (geinotype S hs or ShSh) of Primula vulgaris. These frequencies are the result of a mixture of self-fertilization of the parent homostyle and cross-fertilization with other homostyles, the cross pin X homostyle being illegitimate (Bodmer [1958]). Homostyles giving rise to segregating progenies must be of the genotype Shs. Hence the expected proportion of pins among progenies which are the result of self-fertilization only is 4, and among those which are the result of cross-fertilization only is -6 The overall proportion may vary from A to 6 according to the amount of cross-fertilization and differences in the observed proportioni of pins for different years will indicate differences in the amount of cross-fertilization in these years. The nine observed proportions for the years 1946-55 are given in Table 1. The overall X2 for deviation from a 1:3 ratio is highly significant (p < 0.1%,) showing that cross-fertilization did occur. The