Analysing Discrete Data

Case 1: Chisquare Test for Independence

Example 1 : Rogain
Classification 1: Group (Treatment or Control)
Classification 2: Amount of Hair Growth (No Growth, New Vellus, Minimal Growth, Moderate Growth, Dense Growth)

Example 2: Drowning in Los Angeles:
Classification 1: Gender (Male or Female)
Classification 2: Method of Drowning ( Swimming Pool, Ocean ect.)

Basic Question: Is there a connection between the classifications? (Or: are they independent?)

Step 1: Graph (Multiple Barchart)

Example 1 : Rogain

Commands: open worksheet rogaine.mtw (data in the form of a table)
Graph > Barchart, Values from a table, Cluster, ok
Graph variables c2-c6, Row labels: Groups
Bar Chart options > Show Y as Percent, Within categories of level 1

Example 2 : Drowning
The graph has to based on percentages, but the graphs MINITAB does on its own are no good. Here is what you need to do:
Calc > Calculator, Store in %Male, Expression: ROUND('Male' / SUM('Male') * 100,1)
Calc > Calculator, Store in %Female, Expression: ROUND('Female' / SUM('Female') * 100,1)
Graph > Barchart, Values from a table, Cluster, ok
Graph variables %Male %Female, Row labels: Methods
Bar Chart options > Decreasing Y


Often (mostly?) you should use percentages rather than counts.

Step 2: Hypothesis Test
H₀: Classifications are independent
H_a: Classifications are dependent

Example Rogaine:
H₀: Classifications are independent = Rogaine does not work
H_a: Classifications are dependent = Rogaine does work

Example Drowning
H₀: Classifications are independent = there is no difference in the method of drowning between men and women.
H_a: Classifications are dependent = there is some difference in the method of drowning between men and women.

Stat - Tables - Chisquare Test - Columns containing the table (remember you need to use the counts here, not percentages)
Printout has p value

Just like the other methods we talked about in this class, the Chisquare test for Independence also has its assumptions. They are:
• no more than 20% of the expected cell counts less than 5
• all expected cell counts greater than 1

If there is a problem, join classifications

Case 2: Chisquare Goodness-of-Fit Test

Example 1 : Seat belt use
Column "Observed" with the counts for each type of injury
Column "Historic Percentages" with the theory to be tested
Make column 'Expected' by Calc >Calculator, Store in Expected, Expression: ROUND(SUM('Observed') * 'Historical Percentages' / 100,1)

Example 2: Gregor Mendel's pea experiment
Column "Observed"
Theory: counts of peas should have ratios 9:3:3:1
Note 9+3+3+1=16, so 9/16^th of the peas should be smooth yellow, 3/16^th wrinkled yellow and so on.
To make column 'Expected' do this:
Make column 'Proportions' with numbers 9, 3, 3,1 use calculator to devide by 16 (=9+3+3+1)
Make column 'Expected' by Calc >Calculator, Store in Expected, Expression: ROUND(SUM('Observed') * 'Proportions',1)

Graphs: once you have columns 'Observed' and 'Expected' you can do the multiple barchart for them. Note: here you do not need to use percentages.

Hypothesis Testing:
H₀: Data agrees with Theory
H_a: Data does not agree with Theory

Example For seat belt theory is that historical percentages are still correct, that is, seat belts do not make a difference.
H₀: Data agrees with Theory (Seat belts do not work)
H_a: Data does not agree with Theory (Seat belts do work)

Example For Mendels pea data theory is that counts have (roughly) ratios 9:3:3:1
H₀: Data agrees with Theory (Ratios are 9:3:3:1)
H_a: Data does not agree with Theory (Ratios are not 9:3:3:1)

Make a new column, called T, with Calc - Calculator - Expression
SUM(('Observed' - 'Expected')**2 / 'Expected')
Calc - Probability Distributions - Chi Square
Input Column: T
Degrees of Freedom: #of classifications - 1
p value = 1 - P( X <= x )

The use of the chisquare goodness of fit test in detecting cheating
Suppose two students take a multiple choice test. Their answersheets are as follows:

Because there are always 4 wrong answers we would expect about 25% of the wrong answers to match. If the two students have a much higher rate of matching wrong answers this might indicate cheating.