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Desipramine is an antidepressant affecting the brain chemicals that may become unbalanced and cause depression. It was tested for recovery from cocaine addiction. Treatment with desipramine was compared to a standard treatment (lithium, with strong anti-manic effects) and a placebo. Is desipramine effective in preventing relapses?

Hypothesis: no association

We use the chi-square () test to assess the null

hypothesis of no relationship between the two

categorical variables of a two-way table.

Two-way tables sort the data according to two

categorical variables. We want to test the

hypothesis that there is no relationship between

these two categorical variables (H0).

Expected Cell Counts

To test this hypothesis, we compare actual counts from the sample data with expected counts, given the null hypothesis of no relationship.

The expected count in any cell of a two-way table when H0 is true is:

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25*26/74 ≈ 8.7825*0.35

16.2225*0.65

9.1426*0.35

16.8625*0.65

8.0823*0.35

14.9225*0.65

Desipramine

Lithium

Placebo

Expected relapse counts

No Yes35% 35%35%

Overall non-relapse %

Observed

Cocaine addiction

NOTE: 26/74 = 0.35

15/25=.6

7/26=.274/23=.17

Desipramine is an antidepressant affecting the brain chemicals that may become unbalanced and cause depression. It was thus tested for recovery from cocaine addiction. Treatment with desipramine was compared to a standard treatment (lithium, with strong anti-manic effects) and a placebo.

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The Chi-Square Statistic

To see if the data give convincing evidence against the null hypothesis, we compare the observed counts from our sample with the expected counts assuming H0 is true.

The test statistic that makes the comparison is the chi-square statistic.

The chi-square statistic is a measure of how far the observed counts are from the expected counts. The formula for the statistic is:

Where “observed count” represents an observed cell count, “expected count” represents the expected count for the same cell, and the sum is over all r c cells in the table.

The chi-square statistic is a measure of how far the observed counts are from the expected counts. The formula for the statistic is:

Where “observed count” represents an observed cell count, “expected count” represents the expected count for the same cell, and the sum is over all r c cells in the table.

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The Chi-Square DistributionsIf the expected counts are large and the observed counts are very different, a large value of 2 will result, providing evidence against the null hypothesis. The P-value for a 2 test comes from comparing the value of the 2 statistic with critical values for a chi-square distribution.

The chi-square distributions are a family of distributions that take only positive values and are skewed to the right. A particular 2 distribution is specified by giving its degrees of freedom.

The 2 test for a two-way table with r rows and c columns uses critical values from the 2 distribution with (r – 1)(c – 1) degrees of freedom. The P-value is the area under the density curve of this 2 distribution to the right of the value of the test statistic.

The chi-square distributions are a family of distributions that take only positive values and are skewed to the right. A particular 2 distribution is specified by giving its degrees of freedom.

The 2 test for a two-way table with r rows and c columns uses critical values from the 2 distribution with (r – 1)(c – 1) degrees of freedom. The P-value is the area under the density curve of this 2 distribution to the right of the value of the test statistic.

The Chi-Square DistributionsThe Chi-Square Distributions

When is it safe to use a 2 test?

We can safely use the chi-square test when:

The samples are simple random samples (SRS).

All individual expected counts are 1 or more (≥1)

No more than 20% of expected counts are less than 5 (< 5)

For a 2x2 table, this implies that all four expected counts

should be 5 or more.

ObservedCocaine addiction

The p-value is <0.005 or half a

percent. This is very significant.

We reject the null hypothesis of

no association and conclude that

there is a significant relationship

between treatment (desipramine,

lithium, placebo) and outcome

(relapse or not).

Try using Software. Statistical

software output for the cocaine study

- look at the last lines of output:

HW: Read section 9.1; do #9.1, 9.2, 9.7, 9.15, 9.16, 9.19, 9.20, 9.28, 9.36(a)

Tests

N DF -LogLike RSquare (U)

74 2 5.375719 0.1121

Test ChiSquare

Prob>ChiSqLikelihood

Ratio10.751 0.0046*

Pearson 10.729 0.0047*

Summary: Analyzing two-way tables

When analyzing relationships between two categorical variables, follow this procedure:

1. Calculate descriptive statistics that convey the important information in the table—usually column or row percents. Always take percents with respect to the explanatory variable's totals!

2. Find the expected counts and use them to compute the X2 statistic.

3. Compare your X2 statistic to the chi-square critical values from Table F to find the approximate P-value for your test - or use software to do all the computations, including the expected values!

4. Draw a conclusion about the association between the row and column variables - don't forget the context!

Table F gives upper

critical values for many

χ2 distributions.

Finding the p-value with Table F

The χ2 distributions are a family of distributions that can take only

positive values, are skewed to the right, and are described by a specific

degrees of freedom.

Table F

df = (r−1)(c−1)

If 2 =

10.73, the

p-value is

between

0.0025 and

0.005

From

software

we have

P=.0047

In a 3x2table, df=2x1=2