Logistic Regression: A Self-learning Text, Third Edition (Statistics in the Health Sciences)

Wald statistic (for largen):

Z¼

^b

sb^is approximatelyN(0, 1)

or

Z^2 is approximatelyw^2 with 1 df

Variable

ML

Coefficient S.E.

Chi

sq P

X 1 b^ 1 s^b 1 w^2 P







Xj ^bj s^bj w^2 P







Xk b^k s^bk w^2 P

LRZWald^2 in large samples

LR 6 ¼ZWald^2 in small to moderate
samples

LR preferred (statistical)

Wald convenient – fit only one
model

The Wald test statistic is computed by dividing

the estimated coefficient of interest by its stan-

dard error. This test statistic has approxi-

mately a normal (0, 1), orZ, distribution in

large samples. The square of thisZstatistic is

approximately a chi-square statistic with one

degree of freedom.

In carrying out the Wald test, the information

required is usually provided in the output,

which lists each variable in the model followed

by its ML coefficient and its standard error.

Several packages also compute the chisquare

statistic and aP-value.

When using the listed output, the user must

find the row corresponding to the variable of

interest and either compute the ratio of the

estimated coefficient divided by its standard

error or read off the chi-square statistic and

its correspondingP-value from the output.

The likelihood ratio statistic and its corre-

sponding squared Wald statistic give approxi-

mately the same value in very large samples; so

if one’s study is large enough, it will not matter

which statistic is used.

Nevertheless, in small to moderate samples,

the two statistics may give very different

results. Statisticians have shown that the likeli-

hood ratio statistic is better than the Wald sta-

tistic in such situations. So, when in doubt, it is

recommended that the likelihood ratio statistic

be used. However, the Wald statistic is some-

what convenient to use because only one

model, the full model, needs to be fit.

As an example of a Wald test, consider again

the comparison of Models 1 and 2 described

above. The Wald test for testing the null

hypothesis thatb 3 equals 0 is given by theZ

statistic equal tob^ 3 divided by the standard

error ofb^ 3. The computedZcan be compared

with percentage points from a standard normal

table.

EXAMPLE

Model1: logitP 1 (X)¼aþb 1 X 1 þb 2 X 2

Model2: logitP 2 (X)¼aþb 1 X 1 þb 2 X 2

þb 3 X 3

H 0 : b 3 ¼ 0

Z¼

b^ 3

s^b 3 is approximatelyN(0, 1)

Presentation: V. The Wald Test 139