Chi-Squared Analysis Stickrath

Chi-Squared Analysis Suppose I bet you $1,000 that I can predict whether heads or tails will turn up each time you flip a coin. The first time I say, “heads” you flip the coin and it is heads. I got lucky The second time I say, “heads” you flip the coin and it is heads

Chi-Squared Analysis The third time, fourth time, fifth time, sixth time, seven time, eighth time, and so on I predict heads. Each time you flip heads. At what point do you suspect that I am using a two-headed coin? When do you stop chalking it up to chance and accuse me of using a two-headed coin? You can use statistics to back up your accusations and save yourself $1,000

Chi-Squared Analysis Start with the assumption (null-hypothesis) that the results of the coin flip are due to chance It is easier to disprove something than to prove it You will attempt to disprove your null-hypothesis By showing that it is NOT due to chance you can accuse me of cheating

Chi-Square Test Heads 20 10 100 Tails -10 Comparison of observed results and expected results Null-hypothesis: It is purely due to chance X2 value = Sum of (Observed – Expected)2 Expected X2 value = 20 Categories Observed Expected (O-E) (O-E)2 (O-E)2/E Heads 20 10 100 Tails -10

What if we do a second experiment with a new coin and obtain the results below Null-hypothesis: It is purely due to chance X2 value = Sum of (Observed – Expected)2 Expected X2 value = 0.2 Categories Observed Expected (O-E) (O-E)2 (O-E)2/E Heads 11 10 1 0.1 Tails 9 -1

What conclusion would you make from the data for the two coins? Which data is legitimately due to chance, and which data is not due to chance? In the case of the first coin (two-headed) the chi-squared (X2) value is 20 In the case of the second coin (regular) the chi-squared (X2) value is 0.2 So, the higher the (X2) value…the _______ likely the results are due to chance The lower the (X2) value…the _______ likely the results are due to chance

How low is low enough? The null-hypothesis is that your results are due to chance You are attempting to disprove the null-hypothesis It is easier to disprove something than to prove it How can chi-squared (X2) analysis be used to disprove the null-hypothesis There’s an app for that (actually a chart) To follow the chart you must know two things Degrees of Freedom p-value

Degrees of Freedom The number of values in the final calculation of a statistic that are free to change Let’s say I give you 4 numbers and tell you that they must add up to 100. In addition, I tell you that one of the numbers is 50. The three remaining numbers could be a variety of values as long as the overall total is 100 Choice 1 Choice 2 Choice 3 Number 1 = 50 Number 1 = 50 Number 1 = 50 Number 2 = 30 Number 2 = 5 Number 2 = ? Number 3 = 10 Number 3 = 25 Number 3 = ? Number 4 = 10 Number 4 = 20 Number 4 = ?. There are many more choices that fulfill the conditions

Degrees of Freedom In the example above you have 4 options, one of which is a fixed value (50) 3 numbers are free to change 3 degrees of freedom What if I said you have 5 options, one of which is a fixed value (50) 4 numbers are free to change 4 degrees of freedom The more options you have, the more degrees of freedom you have Generally, in biology degrees of freedom = # categories -1

p-value The null-hypothesis is that your results are due to chance p-value: probability that the null-hypothesis is valid (true) High p-value means null-hypothesis is true Low p-value means that the null-hypothesis is untrue How low is low enough? The significant p-value is 0.05 (5%) A p-value less than 0.05 means that it is less than 5% likely that the results are due to chance A p-value greater than 0.05 means that it is more than 5% likely that the results are due to chance

Two-headed coin Big X2 = Small p-value = Not due to chance = Statistically Significant Data The X2 value for our two-headed coin was 20 The number of options were 2 (heads or tails) = 1 degree of freedom The significant p-value is always 0.05 or less Critical value for 1 degree of freedom is 3.84 20 is greater than 3.84 so p-value is less than 0.05 20 = 0.000001 = Not due to chance = Statistically significant

Regular coin Small X2 = Large p-value = Due to chance = Statistically Insignificant Data The X2 value for our two-headed coin was 0.2 The number of options were 2 (heads or tails) = 1 degree of freedom The significant p-value is always 0.05 or less Critical value for 1 degree of freedom is 3.84 0.2 is lower than 3.84 so p-value is more than 0.05 0.2 = 0.65 = Due to chance = Statistically insignificant

Simple vs. Complex In the case of the two-headed coin, you have simple expectations 50:50 heads to tails What about more complex problems?

Teaching Example 100 students took my exam X2 =11.93 Degrees of Freedom = # categories -1 = 5-1 = 4 Categories Observed Expected (O-E) (O-E)2 (O-E)2/E A 20 15 5 25 1.67 B 22 -3 9 0.36 C 35 40 -5 0.63 D 23 8 64 4.27 F

Did my student meet my expectations? Degrees of Freedom = # categories -1 = 5-1 = 4