Chi Square Tutorial

8/19/2019 Chi Square Tutorial

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TUTORIAL 1 – CHI SQUARE TEST

Introduction

Inheritance obeys the same rules of probablilty that apply to tossing coins androlling dice. Mendel’s great achievement was his recognition from experimentalresults that this is so.

A simple case: given a pair of alleles of a gene, one dominant and one recessive,their recombination at fertilization is like flipping two coins at the same time.

A coin has two sides, head and tail (like two alleles of a gene). If you flip two coinsand examine the paired outcomes, you will observe three possible combinations:HH, HT, and TT. Since there is half a chance that either coin will come up heads,the probability of a homozygous outcome, HH (also of hh), is 1/2 x 1/2 = 1/4. Theoutcome, HT, can arise in two ways so its probability is 1/4 + 1/4 = 2/4. Theoutcome of many such trials of HT x HT (flipping 2 coins and examining whichpair of faces is up) will be 1/4HH + 2/4HT + 1/4 TT. The more times you flip the

i t th th l ill t th id l ti



coins together the closer you will come to these ideal ratios

Example

Suppose you counted 79 R_ and 33 rr. The total number of individuals you

counted, N, is 112. You expect3

/4 to be R_ (84) and1

/4 to be rr (28). Are yourresults close enough to these ratios for you to accept the null hypothesis — thatthere is no real difference? The Chi-square test is one tool for making thisdecision.

Phenotypes Observed(O)

Expected(E)

D = O - E D2 D2/E

R_ 79 (3/4) x 112 = 84 -5 25 0.30

rr 33 (1/4) x 112 = 28 5 25 0.89

Total 112 112 0 1.19

Χ2 = ∑ (Observed - Expected)2/(Expected).

This means add up the values in the last column.

You can compare the chi-square sum, 1.19, with the numbers in a table ofcritical values to decide whether to accept the null hypothesis — that the observedresults are so close to expected results that there is no difference, and ouroriginal hypothesis is accepted.

Table 1. Selected percentile values of the Χ2 distribution

Probablilities



Mendelian Data

Question 1 - Monohybrid cross

In corn genetics, P (purple) is the dominant allele, and p (yellow) is the recessive

allele. Wild-type plants have yellow-pigmented grains, while mutant individuals have

purple grains. The outcome of a monohybrid cross of two parent plants (Pp X pp)

yielded 39 purple kernels from a total of 110 kernels counted. Answer the following

question, with reference to a cross-breeding diagram (Branch or Punnet Square).

Phenotypes Observed(O)

Expected(E)

D = O - E D2 D2/E

Total 0

Χ2 : …………………….

Degrees of freedom : …………………….

Range of probability : …………………….

A t j t ll h th i



CHI SQUARE TABLES

df\area 0.995 0.990 0.975 0.950 0.900 0.750 0.500 0.250 0.100 0.050 0.025 0.010 0.005

1 0.00004 0.00016 0.00098 0.00393 0.01579 0.10153 0.45494 1.32330 2.70554 3.84146 5.02389 6.63490 7.87944

2 0.01003 0.02010 0.05064 0.10259 0.21072 0.57536 1.38629 2.77259 4.60517 5.99146 7.37776 9.21034 10.59663

3 0.07172 0.11483 0.21580 0.35185 0.58437 1.21253 2.36597 4.10834 6.25139 7.81473 9.34840 11.34487 12.83816

4 0.20699 0.29711 0.48442 0.71072 1.06362 1.92256 3.35669 5.38527 7.77944 9.48773 11.14329 13.27670 14.86026

5 0.41174 0.55430 0.83121 1.14548 1.61031 2.67460 4.35146 6.62568 9.23636 11.07050 12.83250 15.08627 16.74960

Chi squared

Degrees of freedom (df)

10 9 8 7 6 5 4 3 2 1 p value

2.56 2.09 1.65 1.24 0.87 0.55 0.30 0.11 0.02 0.00 .99

4.87 4.17 3.49 2.83 2.20 1.61 1.06 0.58 0.21 0.02 .90

6.18 5.38 4.59 3.82 3.07 2.34 1.65 1.01 0.45 0.06 .807.27 6.39 5.53 4.67 3.83 3.00 2.19 1.42 0.71 0.15 .70

8.30 7.36 6.42 5.49 4.57 3.66 2.75 1.87 1.02 0.27 .60

9.34 8.34 7.34 6.35 5.35 4.35 3.36 2.37 1.39 0.45 .50

10.47 9.41 8.35 7.28 6.21 5.13 4.04 2.95 1.83 0.71 .40

11.78 10.66 9.52 8.38 7.23 6.06 4.88 3.66 2.41 1.07 .30

13.44 12.24 11.03 9.80 8.56 7.29 5.99 4.64 3.22 1.64 .20

14.53 13.29 12.03 10.75 9.45 8.12 6.74 5.32 3.79 2.07 .15

15.99 14.68 13.36 12.02 10.64 9.24 7.78 6.25 4.61 2.71 .10

16.35 15.03 13.70 12.34 10.95 9.52 8.04 6.49 4.82 2.87 .09

16.75 15.42 14.07 12.69 11.28 9.84 8.34 6.76 5.05 3.06 .08

17.20 15.85 14.48 13.09 11.66 10.19 8.67 7.06 5.32 3.28 .0717.71 16.35 14.96 13.54 12.09 10.60 9.04 7.41 5.63 3.54 .06

18.31 16.92 15.51 14.07 12.59 11.07 9.49 7.81 5.99 3.84 .05

19.02 17.61 16.17 14.70 13.20 11.64 10.03 8.31 6.44 4.22 .04

19.92 18.48 17.01 15.51 13.97 12.37 10.71 8.95 7.01 4.71 .03

21.16 19.68 18.17 16.62 15.03 13.39 11.67 9.84 7.82 5.41 .02

23.21 21.67 20.09 18.48 16.81 15.09 13.28 11.34 9.21 6.63 .01

29.59 27.88 26.12 24.32 22.46 20.52 18.47 16.27 13.82 10.83 .001

Chi Square Tutorial

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