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Background

I am reading the textbook Applied Logistic Regression by David Hosmer, specifically chapter 4, which discusses logistic regression model assesment of fit. Hosmer gives an interesting example of how classification tables may not imply goodness of fit for a logistic regression model.

Hosmer introduces univariate logistic regression as follows:

Given a dichotomous random variable $Y\in\{0,1\}$, we model the expected value of $Y$ given $X$ using a logistic function.

$$E(Y|X) = \pi(x) = \frac{1}{1+e^{-(\beta_0 + \beta_1X)}}$$

and he goes on to define the logit and derive the likelihood function, but I don't think those are needed for the problem below.

Problem

Suppose that $P(Y=1) = \theta_1$ and $X \sim N(0,1)$ in the group with $Y=0$ and $X \sim N(\mu,1)$ in the group with $Y=1$.

(1) It can be shown that the slope coefficient for the logistic regression model is $\beta_1 = \mu$, and the intercept is

$$ \beta_0 = \ln\left[\frac{\theta_1}{1-\theta_1}\right] - \frac{\mu^2}{2} $$

(2) Then, the probability of misclassification (PMC), may be shown to be

$$ \mathrm{PMC} = \theta_1 \Phi\left[\frac{1}{\beta_1}\ln\left(\frac{1-\theta_1}{\theta_1}\right)-\frac{\beta_1}{2}\right] + (1-\theta_1)\Phi\left[\frac{1}{\beta_1}\ln\left(\frac{\theta_1}{1-\theta_1}\right)-\frac{\beta_1}{2}\right] $$

where $\Phi$ is the standard normal CDF. This implies that the misclassification rate depends on the slope of the model rather than the goodness of fit.

Discussion

Hosmer says that (1) is easy to show, and is shown in an old textbook named Discriminant Analysis by Lachenbruch. I was unable to find a PDF of this textbook. I think to show (1), I should use conditional probability somehow, but I'm not sure. I assume that (2) will follow using similar techniques.

How can I show (1) and (2)?

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    $\begingroup$ It's been a while, but the B_0 equation is just a manipulation of the first equation you have (E(Y|X)=). Just solving that equation for B_0 I believe. $\endgroup$ Apr 3 at 15:38
  • $\begingroup$ Any ideas on how we reason that $\beta_1=\mu$? $\endgroup$ Apr 3 at 15:51
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    $\begingroup$ Honestly I haven't attacked the problem from this angle before. I would check out this link though, it gives a pretty thorough formulation of Logistic Regression. sciencedirect.com/topics/mathematics/logit-link-function $\endgroup$ Apr 3 at 21:59
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This notation is not perfect, but it helps keep the conditioning straight:

$$E(Y|X=x) = \frac{1}{1+e^{-(\beta_0 + \beta_1 x)}}$$

Since $X \sim N(0,1)$ and $N(\mu, 1)$,

$$f(X|Y=1) = \frac{1}{\sqrt{2 \pi} \sigma} e^{\frac{-(x-\mu)^2}{2 \sigma}} = \frac{1}{\sqrt{2 \pi}} e^{\frac{-(x-\mu)^2}{2}}$$

$$f(X|Y=0) = \frac{1}{\sqrt{2 \pi}} e^{\frac{-x^2}{2}}$$

Recognize that

$$f(X) = P(Y=0)f(X|Y=0) + P(Y=1)f(X|Y=1)$$

$$E(Y|X=x) = 0 P(Y = 0|X=x) + 1 P(Y=1|X=x) = P(Y=1|X=x)$$

And using Bayes rule

$$ = \frac{f(X|Y=1) P(Y=1)}{f(X)} = \frac{\theta \frac{1}{\sqrt{2 \pi}} e^{\frac{-(x-\mu)^2}{2}}}{\theta \frac{1}{\sqrt{2 \pi}} e^{\frac{-(x-\mu)^2}{2}} + (1-\theta) \frac{1}{\sqrt{2 \pi}} e^{\frac{-x^2}{2}}}$$

$$ = \frac{1}{1 + \frac{1-\theta}{\theta} e^{-x^2 / 2 + (x-\mu)^2/2}} = \frac{1}{1+e^{ln(\frac{1-\theta}{\theta}) + \mu^2/2 - \mu x}}$$

Therefore $\beta_1 = \mu$ and $\beta_0 = log(\frac{\theta}{1-\theta}) - \mu^2 / 2$

Next the probability of misclassification is when a different class would be predicted than the actual. Utilizing 0.5 as the class threshold,

$$PMC = P(Y=1) P(\frac{1}{1+e^{-(\beta_0 + \beta_1 x)}} < 0.5 | Y=1) + P(Y=0)P(\frac{1}{1+e^{-(\beta_0 + \beta_1 x)}} > 0.5 | Y = 0)$$

Note that $\frac{1}{1+e^{-(\beta_0 + \beta_1 x)}} = 1/2$ then algebra shows $x = \frac{-\beta_0}{\beta_1}$

Now we can write the PMC:

$$PMC = P(Y=1)P(X < \frac{1}{\beta_1} log(\frac{1-\theta}{\theta}) + \beta_1 / 2 < 0.5|Y=1) + P(Y=0)P(X < \frac{1}{\beta_1} log(\frac{1-\theta}{\theta}) + \beta_1 / 2 > 0.5 | Y = 0)$$

Now, change the $P(X < *)$ to be a statement for a normal CDF

$$ = \theta \Phi(\frac{1}{\beta_1} log(\frac{1-\theta}{\theta}) + \beta_1 / 2 - \mu) + (1-\theta)[1-\Phi(\frac{1}{\beta_1} log(\frac{1-\theta}{\theta}) + \beta_1 / 2)]$$

$$ = \theta \Phi(\frac{1}{\beta_1} log(\frac{1-\theta}{\theta}) - \beta_1 / 2) + (1-\theta)[1-\Phi(\frac{1}{\beta_1} log(\frac{1-\theta}{\theta}) + \beta_1 / 2)]$$

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  • $\begingroup$ Thanks for your answer! It's been really helpful to me. I was wondering if you have any comment on how this relates to the goodness of fit of a logistic model? I'm still learning goodness-of-fit metrics for logistic models. Right now, I'm confused about Hosmer's point. It seems like if $\beta_1=\mu$ is increased, PMC will decrease and the fit statistic will increase (so the $p$-value decreases). This makes it seems like misclassification is related to goodness of fit. What do you think? Is there an example where decreasing MCR does not increase goodness of fit? $\endgroup$ Apr 12 at 19:55
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    $\begingroup$ If you interpret goodness of fit to be related to the regression residual variance, then what Hosmer is saying is that the residual variance doesn't show up in the calculation of PMC at all. The PMC is essentially related to how much overlap there is between the classes along X. If the classes are separate, there will be a large slope and little misclassification. If the classes have overlapping regions, then there will be a low slope and more misclassification. $\endgroup$
    – R Carnell
    Apr 12 at 21:01
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Many thanks to R Carnell for providing the derivation. To increase my understanding, I ran some simulations to validate the theoretical results.

First some helper functions.

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm
from scipy.stats import bernoulli

# expectation of Y given X -- logistic model
def EYX(x,beta0,beta1):
    denom = 1+np.exp(-(beta0 + beta1*x))
    return 1./denom

# predict class of Y given X, using logistic model
def classify(x,beta0,beta1,c):
    return np.asarray([int(EYX(xval,beta0,beta1)>c) for xval in x])

Now simulate 100 observations of the dichotomous variable $Y$ such that $P(Y=1)=\theta_1$ and $X\sim N(\mu,1)$ in the group with $Y=1$ and $X\sim N(0,1)$ in the group with $Y=0$.

# parameters
mu=1
theta=0.7

# classification cutoff
c=0.5

# generate the data using mu,theta
n=100
y=np.sort(bernoulli.rvs(theta,size=n))
x0=norm.rvs(loc= 0,scale=1,size=np.sum(y==0))
x1=norm.rvs(loc=mu,scale=1,size=np.sum(y==1))
x=np.concatenate((x0,x1))

Now we calculate the theoretical model coefficients, predict the class of each $Y_i$

# true beta0,beta1
beta0=np.log(theta/(1-theta)) - 0.5*mu**2
beta1=mu

# predict class using logistic model
yhat=classify(x,beta0,beta1,c)

Below, I plot the data, the logistic model, the predicted classes, and the undelrying distributions for $X|Y=0$ and $X|Y=1$.

plt.plot(x,y,'o',label='data');
xv = np.linspace(-3,3,100);
plt.plot(xv,(1-theta)*norm.pdf(xv),'k--',label='f(X|Y=0)');
plt.plot(xv,theta*norm.pdf(xv,loc=mu),'k--',label='f(X|Y=1)');
plt.plot(xv,EYX(xv,beta0,beta1),label='E(Y|X)');
plt.plot(xv,classify(xv,beta0,beta1,c),label='Predicted Class')
plt.legend();

enter image description here

We can also compute the theoretical misclassification rate.

def MCR(x,y,beta0,beta1,c):
    yhat = classify(x,beta0,beta1,c)
    num_misclass = float(np.sum(np.abs(yhat-y)>0))
    return num_misclass/float(y.size)
    
true_PMC = theta*norm.cdf(np.log((1-theta)/theta)/beta1 - beta1/2.) \
        + (1-theta)*norm.cdf(np.log(theta/(1-theta))/beta1 - beta1/2.)

obsv_PMC = MCR(x,y,beta0,beta1,c)

print(true_PMC,obsv_PMC)

which yields

0.2530043786236347 0.21

Does the observed misclassification rate match the theoretical value, on average? A quick simulation confirms.

ntimes=2000
obsv_PMC=np.zeros(ntimes)

for i in range(ntimes):
    y=np.sort(bernoulli.rvs(theta,size=n))
    x0=norm.rvs(loc= 0,scale=1,size=np.sum(y==0))
    x1=norm.rvs(loc=mu,scale=1,size=np.sum(y==1))
    x=np.concatenate((x0,x1))
    obsv_PMC[i] = MCR(x,y,beta0,beta1,c)

plt.hist(obsv_PMC,bins=100,alpha=0.7);
plt.axvline(x=true_PMC,
            linestyle='-',
            color='r',
            label='True PMC');
plt.legend();
plt.title('Misclassification Rate');

enter image description here

The only confusing aspect is exactly how this relates to the goodness of fit. How does this result imply that there exist cases where PMC goes to zero but goodness of fit does not increase?

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