Quantitative Methods

for Lawyers

Chi Square ( ) Distribution

Class #12

χ 2

( ad - bc)2 (a + b + c +d)

( a + b) (c +d) (b +d) ( a + c)

Chi Square ( ) Test χ 2

@ computational

computationallegalstudies.com

professor daniel martin katz danielmartinkatz.com

lexpredict.com slideshare.net/DanielKatz

Generally - Two types of random variables numerical and categorical

“What was your college major ?” or “Do you own a bike?” are categorical because they yield data such as “Economics” or “no.”

Categorical versus Numerical Data

categorical variables yield data in categories numerical variables yield data in numerical form

Categorical variables

“How tall are you?” or “What is your G.P.A.?” are numerical. Numerical data can be either discrete or continuous.

Numerical variables

A chi square (χ 2) statistic is used to investigate whether distributions of categorical variables differ from one another

Chi Square ( ) Statistic χ 2

The Chi Square statistic compares the tallies or counts of categorical responses between two (or more) independent groups.

(note: Chi square tests can only be used on actual numbers and not on percentages, proportions, means, etc.)

There are several types of chi square tests depending on the way the data was collected and the hypothesis being tested.

Imagine the simplest case of a 2 x 2 contingency table

If we set the 2 x 2 table to the general notation shown below in Table 1, using the letters a, b, c, and d to denote the contents of the cells:

Variable 2 Data Type 1 Data Type 2 Totals

Category 1 a b a + b

Category 2 c d c + d

Total a + c b + d a+b+c+d =N

Chi Square ( ) Statistic χ 2

Variable 2 Data Type 1 Data Type 2 Totals

Category 1 a b a + b

Category 2 c d c + d

Total a + c b + d a+b+c+d =N

χ 2 = ( ad - bc)2 (a + b + c +d)( a + b) (c +d) (b +d) ( a + c)

Note: notice that the four components of the denominator are the four totals from the table columns and rows

Chi Square ( ) Statistic χ 2

Male Female Totals

Not Research Asst 319 323 642

Research Assistant 60 34 94

Total 379 357 736

In Our Prior Class We Discussed Hypothesis Testing and that is the approach we would like to use here

Chi Square is a technique to consider whether the observed gender disparity in RA positions is too large to be result of chance

Chi Square ( ) Statistic χ 2

Male Female Totals

Not Research Asst 319 323 642

Research Assistant 60 34 94

Total 379 357 736

Ho is the Null Hypothesis

H1 is the Alternative Hypothesis

In this Case, Please Describe Each of these in simple words

Chi Square ( ) Statistic χ 2

Male Female Totals

Not Research Asst 319 323 642

Research Assistant 60 34 94

Total 379 357 736

Ho: Gender Does Not Affect Probability of Being RA

Chi Square ( ) Statistic χ 2

Male Female Totals

Not Research Asst 319 323 642

Research Assistant 60 34 94

Total 379 357 736

We need to understand the Expected Value for this question as it sets our baseline expectations

Chi Square ( ) Statistic χ 2

Male Female Totals

Not Research Asst 319 323 642

Research Assistant 60 34 94

Total 379 357 736

Male Female Totals

Not Research Asst 330.6 311.4 642

Research Assistant 48.4 45.6 94

Total 379 357 736

Chi Square ( ) Statistic χ 2

We need to understand the Expected Value for this question as it sets our baseline expectations

Chi Square is all about comparing expected values to the observed/actual values

Male Female TotalsNot Research Asst 319 323 642Research Assistant 60 34 94Total 379 357 736

Male Female TotalsNot Research Asst 330.6 311.4 642Research Assistant 48.4 45.6 94Total 379 357 736

Chi Square ( ) Statistic χ 2

Male Female

Not Research Asst 0.4 0.4

Research Assistant 2.8 3

Here is the Chi Square Calculation for the Student Population :

= .4 + .4 + 2.8 + 3.0 = 6.6χ 2

Chi Square ( ) Statistic χ 2

What Does a χ 2 value of 6.6 Tell Us?

Need to Look at the P Value on Chi Square Table

Here the Degrees of Freedom are = 1 (more on D.F. later)

Thus , t here i s roughly a 1% probability that the disparity was generated by chance

Chi Square ( ) Statistic χ 2

Suppose you conducted a drug trial on a group and you hypothesize that the group receiving the drug would survive at a rate higher than those that did not receive the drug.

Ho: Group survival is independent of drug treatment

Ha: Group survival is associated with drug treatment

Chi Square ( ) Statistic χ 2

You conduct the study and collect the following data:

χ 2 = ( ad - bc)2 (a + b + c +d)( a + b) (c +d) (b +d) ( a + c)

Chi Square ( ) Statistic χ 2

Applying the formula above we get: Chi square = 105 [(36)(25) - (14)(30) ]2 / (50)(55)(39)(66) = 3.418

Chi Square ( ) Statistic χ 2

How Do We Determine How Many Degrees of Freedom? Before we can proceed we need to know how many degrees of freedom we have. When a comparison is made between one sample and another, a simple rule is that the degrees of freedom equal (number of columns minus one) x (number of rows minus one) not counting the totals for rows or columns. For our data this gives (2-1) x (2-1) = 1.

Chi Square ( ) Statistic χ 2

We now have our chi square statistic (x2 = 3.418), our predetermined alpha level of significance (0.05), and our degrees of freedom (df =1).

Entering the Chi square distribution table with 1 degree of freedom and reading along the row we find our value of x2 (3.418) lies between 2.706 and 3.841.

Chi Square ( ) Statistic χ 2

Male,

CitizenFemale, Citizen

Male, Foreign

Female, Foreign

TOTAL

Research Asst & Grant

12 9 7 9 37

Grant Only 32 62 22 44 160

Research Asst Only

33 11 8 5 57

No Grant & No Research Asst

199 112 66 105 482

TOTAL 276 194 103 163 736

Here We Have Additional Category - Foreign v. Citizen

Suppose You Learn that School Offers Student Grant and that Professor Consider this Before Hiring

Further Assume that Various Immigration Laws Make it more difficult for foreign students to get RA positions

Raw Data Expected Value

Chi Square Calculation

χ 2 = 41.2

χ 2 = 41.2

How Many Degrees of Freedom?

Rule of Thumb is Subtract 1 from # of Rows & Subtract 1 from # of Columns.

Then, Multiply the Result

(4 -1 ) x ( 4 - 1) = 9

χ 2 = 41.2

How Many Degrees of Freedom?

Rule of Thumb is Subtract 1 from # of Rows & Subtract 1 from # of Columns.

Then, Multiply the Result

(4 -1 ) x ( 4 - 1) = 9

41.2 > 27.88

χ 2 = 41.2

How Many Degrees of Freedom?

Rule of Thumb is Subtract 1 from # of Rows & Subtract 1 from # of Columns.

Then, Multiply the Result

(4 -1 ) x ( 4 - 1) = 9

http://sites.stat.psu.edu/~mga/401/tables/Chi-square-table.pdf

Access the Chi Squared Table

http://www.quantpsy.org/chisq/chisq.htm

Calculating the Chi-Square Test

Daniel Martin Katz

@ computational

computationallegalstudies.com

lexpredict.com

danielmartinkatz.com

illinois tech - chicago kent college of law@