logistic regression significance checking

Testing for the Significance of the
Coefficients
 After estimating the coefficients of the logistic
regression model using the maximum likelihood
method, the next step is to assess the statistical
significance of the independent variables.
 This involves formulating and testing a statistical
hypothesis to determine whether the predictors in the
model are significantly related to the outcome
variable.
 The general approach for performing this test is
applicable across different types of models, with the
specific details varying based on the particular model
being used.

 The null hypothesis typically states that the coefficient for
a given independent variable is equal to zero, meaning
that the variable is not significantly associated with the
outcome after accounting for the other predictors in the
model.
 The alternative hypothesis is that the coefficient is not
equal to zero, indicating that the variable has a significant
effect on the outcome.
 To test this hypothesis, we can use the Wald statistic,
which is the ratio of the estimated coefficient to its
standard error.

Under the null hypothesis, the Wald statistic
follows a standard normal distribution, allowing
us to compute a p-value and assess the statistical
significance of the coefficient.
In addition to the Wald statistic, other test
statistics, such as the likelihood ratio test and the
score test, can also be used to evaluate the
significance of the coefficients in a logistic
regression model.

Dichotomous Independent Variables in
Logistic Regression
 This section provides a detailed explanation of how to
interpret logistic regression coefficients when the
independent variable is nominal-scaled and dichotomous
(i.e., measured at two levels).
 When the independent variable, x, is coded as either 0 or 1,
the difference in the logit for a subject with x = 1 and x = 0
is simply the value of the regression coefficient, β1.
 This can be shown through a straightforward algebraic
derivation: g(1) - g(0) = (β0 + β1 × 1) - (β0 + β1 × 0) = (β0
+ β1) - (β0) = β1.

The key steps in this process are:
1) Identifying the coding of the dichotomous
independent variable,
2) Calculating the difference in the logit for x = 1 and x
= 0,
3) Recognizing that this difference is equal to the
regression coefficient β1, and
4) Interpreting the coefficient accordingly.

 The interpretation of logistic regression coefficients when
the independent variable is nominal-scaled and
dichotomous (i.e., measured at two levels) involves a multi-
step process.
 First, we need to identify the two values of the covariate to
be compared, typically coded as 0 and 1.
 Next, we substitute these values into the equation for the
logit, calculating the difference between the logit when x =
1 and the logit when x = 0.
 As shown, for a dichotomous covariate coded 0 and 1, this
difference is equal to the regression coefficient β1.
 This logit difference represents the change in the log of the
odds associated with a one-unit change in the covariate.

 The odds ratio represents the ratio of the odds of the outcome being
present when x = 1 to the odds when x = 0. Importantly, for a
logistic regression model with a dichotomous independent variable
coded 0 and 1, the odds ratio is simply the exponentiated value of
the regression coefficient, OR = eβ1.
 This relationship between the odds ratio and the logistic regression
coefficient is a crucial step in interpreting the effect of a
dichotomous covariate.
 By exponentiating the logit difference, we can provide a meaningful
and easily interpretable measure of the magnitude and direction of
the association between the independent variable and the outcome of
interest.

The Odds Ratio as a Measure of
Association
 The odds ratio is a powerful measure of association in logistic
regression, particularly when the independent variable is dichotomous.
 The odds ratio represents the ratio of the odds of the outcome being
present when x = 1 to the odds when x = 0. Importantly, for a logistic
regression model with a dichotomous independent variable coded 0 and
1, the odds ratio is simply the exponentiated value of the regression
coefficient, OR = eβ1.
 This relationship between the odds ratio and the logistic regression
coefficient is crucial for interpreting the effect of a dichotomous
covariate.
 By exponentiating the logit difference, we can provide a meaningful
and easily interpretable measure of the magnitude and direction of the
association between the independent variable and the outcome of
interest.
 For example, an OR = 2 would indicate that the odds of the outcome
are twice as high for those with x = 1 compared to those with x = 0.

Confidence Interval Estimation for the
Odds Ratio
 The estimator of the odds ratio, denoted as OR, tends to
have a highly skewed distribution, especially with small
sample sizes.
 This is because the range of the odds ratio is between 0
and ∞, with the null value being 1.
 make inferences based on the sampling distribution of
the log-odds ratio, ln(OR) = β1, which is more likely to
follow a normal distribution even with smaller sample
sizes.
 This allows for the construction of confidence intervals
using the familiar formula:
exp[β1 ± z1-α/2 × SE(β1)].

Odds Ratio Estimates by Software
Packages
Many statistical software packages automatically
provide point and confidence interval estimates
based on the exponentiation of each coefficient in
a fitted logistic regression model.
only provide accurate estimates of the odds ratio
in a few special cases, such as when the
independent variable is dichotomous and coded as
0 and 1, and is not involved in any interactions
with other variables.

consider the effect that coding has on computing
the estimator of the odds ratio.
the estimator is given by OR = exp(β₁̂), and this is
correct as long as the independent variable is
coded as 0 or 1 (or any two distinct values).
However, the choice of coding can impact the
interpretation and the calculation of the
confidence interval for the odds ratio

The effect of Coding on the Confidence
Intervals
 The choice of coding for a dichotomous independent
variable can have a significant impact on the
interpretation and calculation of the confidence interval
for the odds ratio.
 when the covariate is coded as 0 and 1, the estimator of
the odds ratio is simply the exponentiated value of the
regression coefficient, OR = exp(β₁̂).
 However, this straightforward relationship does not hold
for other coding schemes.

• The choice of coding (i.e., the values a and b) can influence the
magnitude and interpretation of the odds ratio, as well as the
calculation of the confidence interval.
• For example, coding the covariate as -1 and 1 will result in an odds
ratio estimator of OR = exp(2β₁̂), which is the square of the odds
ratio obtained when the covariate is coded as 0 and 1.
• It is important to note that software packages may not always
provide the correct odds ratio estimates and confidence intervals,
especially when the coding of the independent variable is not the
standard 0 and 1.
• Therefore, it is crucial for the researcher to understand the four-step
process and apply it correctly, regardless of the coding scheme used,
to ensure accurate interpretation of the odds ratio and its associated
uncertainty.

Polychotomous Independent
Variable
 When the independent variable in a logistic regression model is nominal-scaled
and has more than two levels (a polychotomous variable), the interpretation of
the regression coefficients and the odds ratio becomes more complex.
 Unlike the case of a dichotomous covariate, where the odds ratio represents the
comparison of a single group to a reference group, with a polychotomous
variable, we need to make multiple comparisons to fully characterize the effect.
 The key steps for handling a polychotomous independent variable are:
1) Identify the reference group,
2) Create a set of design variables to represent the remaining categories,
3) Interpret the regression coefficients and odds ratios for each comparison to the
reference group.
 This allows us to evaluate how the odds of the outcome differ across the various
levels of the nominal variable.

Careful attention must be paid to the
interpretation, as the odds ratios now represent the
change in odds for each category relative to the
chosen reference.
The choice of reference group can impact the
magnitude and direction of the odds ratios, so it is
important to select the most meaningful or
clinically relevant reference for the research
question at hand.

Assessing the Fit of the Model
 The model building techniques such as hypothesis tests comparing nested
models, are not true assessments of model fit.
 These methods merely compare the fitted values of different models, not
the actual fit of a single model to the data.
 To properly evaluate the goodness of fit, we must consider both summary
measures of the distance between the observed outcomes (y) and the
model-estimated outcomes (ŷ), as well as a thorough examination of the
individual contributions of each data point to these summary measures.
 The key criteria for assessing model fit are:
(1) the summary measures of distance between y and ŷ should be small, and
(2) the contribution of each individual pair of (yi, ŷi) should be unsystematic
and small relative to the error structure of the model.
 This suggests that a comprehensive model fit assessment involves both
global and local evaluations of the discrepancy between the observed and
predicted outcomes.

Model Fit Assessment Techniques
The key components of a comprehensive
model fit assessment approach are:
-Computation and evaluation of overall measures of fit,
such as the Pearson Chi-Square statistic, deviance, or sum-
of-squares, which provide a global indication of how well
the model fits the data.

- Examination of the individual components of the
summary statistics, often through graphical techniques,
to identify any systematic patterns or outliers that may
indicate areas where the model is not adequately fitting
the data.
- Comparison of the observed and fitted values, not in
relation to a smaller model, but in an absolute sense,
considering the fitted model as a representation of the best
possible (saturated) model.

Measures of Goodness of Fit
 When assessing the fit of a model, it is important to
consider various summary measures of goodness of fit.
 These statistics provide a global indication of how well the
model fits the data, but they do not necessarily reveal
information about the individual contributions of each data
point.
 While a small value for these measures can be a good sign,
it does not rule out the possibility of substantial deviations
from fit for some observations. Conversely, a large value is
a clear indication that the model is not adequately capturing
the underlying relationships in the data.
 An important consideration when using these summary
measures is the effect the fitted model has on the degrees of
freedom available for the assessment.

 The term covariate pattern refers to the unique combinations of
covariate values observed in the data. The number of covariate
patterns can impact the degrees of freedom used in the goodness of
fit calculations, as the assessment is based on the fitted values
determined by the covariates in the model, not the total number of
available covariates.
 In logistic regression, where the outcome variable is binary, the
summary measures of model fit differ from those used in linear
regression.
 Unlike linear regression, where the residual is simply the difference
between the observed and fitted values (y - ŷ), the calculation of
residuals in logistic regression must account for the fact that the
fitted values represent estimated probabilities rather than continuous
outcomes.

 The three key summary measures used to assess the goodness of fit
in logistic regression are the Pearson Chi-Square statistic, the
deviance, and the sum-of-squares.
 For a particular covariate pattern j, the Pearson residual is
calculated as:
r(yj, π̂j) = (yj - mjπ̂j) / √(mjπ̂j(1 - π̂j))
 The Pearson Chi-Square statistic is then the sum of the squares of
these Pearson residuals across all covariate patterns:
X2 = Σj=1J [r(yj, π̂j)]2
 The deviance residual and the sum-of-squares residual are two
alternative measures that also capture the discrepancy between the
observed and fitted values. These summary statistics, when
considered alongside the examination of individual residuals,
provide a comprehensive assessment of the model's goodness of fit.

logistic regression significance checking

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