Solution to: QSCI 482: LAB for Topic 4

Here’s another set of data regarding color-shape distribution of green-yellow, round-wrinkled peas.

Under Mendelian genetics, the distribution of the four categories (in the order listed below) should be: 9/16, 3/16, 3/16, 1/16; or equivalently 9:3:3:1.

 

Having alread rejected this null hypothesis at the 0.05 level of significance (observed chi-square test statistic = 8.01, use partitioning to show why we rejected. (Are all categories “off” the null hypothesis? Or only some?) Part of the table is not significantly different from Mendelian predictions; the other part of the table is. The chi-square statistics from the parts of the table should add up to a number close to 8.01, the original chi-square (and the df should add up also). Remember, in subdividing chi-square tables, for df=1 do NOT use the continuity correction.

 

Here's the data that you have:

 

COLOR        OBSERVED COUNT  EXPECTED COUNT   RESIDUAL       STD. RESIDUAL

 

Yellowround                135                  140.625                       -5.625              -0.474

Yellowwrinkled              36                    46.875                       -10.875            -1.588

Greenround                   56                    46.875                       9.125               1.333

Greenwrinkled               23                    15.625                       7.375               1.866

 

Total                            250                  250                              0                      NA

 

Note that there are two negative and two positive standardized residual—a first sign of what should be grouped together.  Furthermore, the two positive ones are relatively close together and larger than zero.  This indicates we should try the following two subgroups:

 

YR,Yw

gR.gW

 

The subtests should preserve the relative proportions under Ho.

 

Subtest 1.  Ho: YR:Yw = 9:3 = 3:1;     P(YR) = 3/4, P(Yw) = ¼

 

Color               Obs     Exp                              (O-E)2/E

YR                   135      171*3/4=128.25          0.355

Yw                   36        171*1/4=42.75                        1.066

Total                171                  171                  1.421<3.841, fail to reject

 

Subtest 2.  Ho: gR:gw = 3:1;  P(gR) = 3/4, P(gw=1/4)

 

Color               Obs                 Exp                              (O-E)2/E

gR                   56        79*3/4=59.25              0.178

gw                   23        79*1/4=19.75              0.535

Total                79                    79                    0.713<3.841, fail to reject

 

Accept these groupings, now perform the “summary” test

 

Ho: YR,Yw:gR,gw = (9+3):(3+1) = 12:4 = 3:1;           P(YR,Yw) = 3/4, P(:gR,gw)=1/4

 

Color                           Obs                             Exp                              (O-E)2/E

YR,Yw             135+36=171                250*3/4=187.5                                    1.452

gR.gW             56+23=79                    250*1/4=62.5                          4.356

Total                250                                          250                              5.808>3.841, Reject

 

Conclusion:  It would seem that given the color of the pea the wrinkled or round trait follows the proportions expected under Mendelian theory (e.g. within yellow or within green).  It may be that it is the color trait that is not sorting in a Mendelian way.