Chi Square

What is it? A statistical test to determine if the actual results are different enough from the predicted results to suggest that the hypothesis is not correct Use the null hypothesis: Any difference between the observed and expected data is due to CHANCE The goal of the Chi Square analysis is to confirm or refute this null hypothesis

Calculating Chi Square Χ2= (O-E)2 E O- observed E- expected

If we use pennies… We expect heads ½ the time and tails ½ the time, so The expected value for heads is 0.5 and the expected value for tails is 0.5 In genetics problems, we use a Punnett square to get our expected values…convert the fractions to decimals

Hypothesis If I flip a coin 300 times, I expect 150 heads, 150 tails for a total of 300

Test the Hypothesis I actually flip the penny 300 times… I observe 162 heads, 138 tails, 300 total

Calculate Chi Square Value For each possible outcome, (O-E)2 E Heads: (O-E)2 (162-150)2 = 144/150 = 0.96 Tails: (O-E)2 (138-150)2 = 144/150 = 0.96

Find Total Chi Square Add Chi Square values together 0.96 + 0.96 = 1.92 All the information is placed on a Chi Square Table Class (of data) Expected Observed (O – E) (O – E)2 (O – E)2/E Heads 150 162 12 144 .96 Tails 138 -12 Total 300   Sum of X2 = 1.92

Degrees of Freedom The number of possible outcomes, minus one. With coins, you have two possible outcomes: heads or tails. There is one degree of freedom

Compare total to a table of critical values Use Chi-Square Distribution table Read across appropriate degrees of freedom row Find the range of the p value- it is unlikely you will be exactly on a p value, but somewhere between two

Critical Values of the X2 Distribution (1.92 with 1 degree of freedom)

What does this mean? So what does that mean? A probability of 0.10 corresponds to a “chance” of 10%; a probability of 0.50 to a “chance” of 50%. This chi-square result means that, if our hypothesis is correct, and we performed exactly this experiment over and over again, 10% or 50% of the time, our results would be at least this far from what we predicted. Or, the probability that we would get results at least as bad as these, even though our hypothesis is correct is between 0.10 and 0.50.

So is the hypothesis supported or not? The usual “level of discrimination” used by investigators is P(X2) = 0.05. Thus, if your chi-square value has a probability of 0.05 or lower, it is very likely (but not certain) that your hypothesis is not supported. If the calculated Chi Square is larger than the critical value, we REJECT the null hypothesis because our data is too different from the expected to be due to chance alone…there must be some other explanation

Practice #1

Hypothesis: White fruit is dominant over yellow fruit P = Ww x ww ½ White; ½ yellow 1:1 .5 white; .5 yellow W w Ww ww

Calculate Expected There are 200 total seeds produced (observed) 200 * .5 = 100 (expect 1 out of 2) 200 * .5 = 100 (expect 1 out of 2) _____ 200

Observed: 110 white 90 yellow

Chi Square (Observed – Expected)2 Expected White: (110-100)2 = 1 Add χ2 together 100 to get total. Yellow: (90-100)2 = 1 Total = 2 100

Degrees of freedom = # of possibilities - 1

What does that mean? 1◦ Freedom X2 value of 2 p value is 0.2-0.1 NOT a significant difference between observed and expected Hypothesis is supported by observed results Due to random chance

Practice #2

Hypothesis: White fruit is dominant over yellow fruit P = Ww x ww ½ White; ½ yellow 1:1 .5 white; .5 yellow W w Ww ww

Calculate Expected There are 2000 total seeds produced (observed) 2000 * .5 = 1000 (expect 1 out of 2) 2000* .5 = 1000 (expect 1 out of 2) _____ 2000

Observed: 1100 white 900 yellow

Chi Square (Observed – Expected)2 Expected White: (1100-1000)2 = 10 1000 Yellow: (900-1000)2 = 10 Total = 20

What does that mean? 1◦ Freedom X2 value of 20 p value is ˂0.01 THERE IS a significant difference between observed and expected Hypothesis is NOT supported by observed results Due to some other factor than random chance

Practice #3 ttWw x TtWw (T= tall t= short; W= wide w= narrow) 3/8 tall, wide 1/8 tall, narrow 3/8 short, wide 1/8 short, narrow tW tw TW TtWW TtWw Tw Ttww ttWW ttWw ttww

There are 462 offspring produced Calculate expected: 462 * 0.375 = 173.25 173 (can’t have ¼ of organism) 462 * 0.375 = 173.25 173 462 * 0.125 = 57.75 58 ____ 462

Observed Of the 462 offspring observed, 158 are tall, wide 64 are tall, narrow 143 are short, wide 97 are short, narrow Calculate the Chi Square!

Calculate Chi Square Tall, wide: (158-173)2 = 1.30 173 Tall, narrow: (64-58)2 = 0.62 58 Short, wide: (143-173)2 = 5.20 Short, narrow: (97-58)2 = 26.22 Total = 33.34

Degrees of freedom = # of possibilities - 1

Analyze 3 degrees of freedom (4 possible phenotypes, less one) Chi- Square value of 33.34 p value is ˂ 0.01 Hypothesis is NOT supported by data Likely due to some factor other than chance