# Linear regression with violated assumptions

I am trying to find out the determinants of cognitive function. The outcome variable is the mini–mental state examination which is a 30 point questionnaire response that has score values from 0 to 30(score values >= 27 indicate normal cognitive function and below 27 indicates some sort of impairment. The explanatory variables are age (continuous) and several categorical variables (sex, education, smoking status and presence of diseases hypertension, diabetes and stroke). In multiple linear regressions I found out that the models have outliers and model check revealed many violations. I have used nonlinear function for age such as squared age, log of age and spline form of age with different degrees of freedom, but the model is still not successful.

1. How should outliers be handles in this situation? Is removing outliers an acceptable option?
2. How could I model categorical variables in this situation? Is there non-linear way of handling?
3. What is the interpretation of age in the two models? Is beta estimate of the models a valid estimate in view of violation of assumption?

Any ideas, tips and suggestions are appreciated

> summary(lm1)

Call:
lm(formula = cogf ~ age + education + smoke + sex + hypert +
stroke + alcohol + diabet, data = df)

Residuals:
Min     1Q Median     3Q    Max
-25.11  -0.66   0.25   1.36  50.52

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)              34.07494    0.67807   50.25  < 2e-16 ***
age                      -0.10909    0.00655  -16.65  < 2e-16 ***
educationhigh school      0.90395    0.17780    5.08  3.9e-07 ***
educationuniversity       0.97544    0.19852    4.91  9.4e-07 ***
smokeformer smoker       -0.15043    0.13742   -1.09   0.2738
smokecurrent smoker      -0.30407    0.19400   -1.57   0.1171
sexwoman                 -0.01764    0.13696   -0.13   0.8975
hypertno                 -0.42188    0.13899   -3.04   0.0024 **
strokeno                  1.20713    0.25854    4.67  3.2e-06 ***
alcohollight to moderate  1.06190    0.14838    7.16  1.0e-12 ***
alcoholheavy drinking     1.07520    0.19720    5.45  5.4e-08 ***
diabet                   -0.02752    0.21129   -0.13   0.8964
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.38 on 3053 degrees of freedom
Multiple R-squared:  0.191, Adjusted R-squared:  0.188
F-statistic: 65.6 on 11 and 3053 DF,  p-value: <2e-16

> summary(lm3)

Call:
lm(formula = cogf ~ ns(age, df = 4) + education + smoke + sex +
hypert + stroke + alcohol + diabet, data = df)

Residuals:
Min     1Q Median     3Q    Max
-23.97  -0.59   0.34   1.06  51.29

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)               26.7627     0.3977   67.29  < 2e-16 ***
ns(age, df = 4)1          -0.5330     0.2698   -1.98    0.048 *
ns(age, df = 4)2          -0.4246     0.3461   -1.23    0.220
ns(age, df = 4)3          -6.0312     0.4501  -13.40  < 2e-16 ***
ns(age, df = 4)4          -9.8537     0.6495  -15.17  < 2e-16 ***
educationhigh school       0.8225     0.1741    4.73  2.4e-06 ***
educationuniversity        1.0076     0.1941    5.19  2.2e-07 ***
smokeformer smoker        -0.1326     0.1343   -0.99    0.324
smokecurrent smoker       -0.3044     0.1896   -1.61    0.109
sexwoman                   0.0179     0.1339    0.13    0.894
hypertno                  -0.2652     0.1365   -1.94    0.052 .
strokeno                   1.2804     0.2529    5.06  4.4e-07 ***
alcohollight to moderate   0.9550     0.1455    6.57  6.1e-11 ***
alcoholheavy drinking      0.9891     0.1931    5.12  3.2e-07 ***
diabet                     0.0565     0.2068    0.27    0.785
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.31 on 3050 degrees of freedom
Multiple R-squared:  0.228, Adjusted R-squared:  0.225
F-statistic: 64.4 on 14 and 3050 DF,  p-value: <2e-16


• I am unable to provide images of model diagnostics. Is there any restriction to upload images? – Fitwe May 11 '15 at 12:06
• Yes, users with low reputation cannot post images. – mpiktas May 11 '15 at 12:10
• Thanks, I share the files on google drive drive.google.com/… – Fitwe May 11 '15 at 12:31
• How to handle outliers depends strongly on whether the outliers are extreme values or mistakes such as miscoded observations or misplaces decimals. The small cluster of extreme outliers probably has some explanation that is not in your model. Are they, for example, mini-mental state results from unconscious patients? For patients with language difficulties? If so then omitting them might be sensible. – Michael Lew May 11 '15 at 23:43
• The results are valid and genuine and they are validated by other diagnostic tests. Those who score 0 are conscious patient who have severe cognitive impairment and the test is conducted on conscious, communicative and reactive individuals. On the other hand 30 indicate individuals with normal cognitive function. Thus the test result is OK , my worry was the statistical model violation. – Fitwe May 12 '15 at 12:42

3. Provided both the age and the outcome are coded in units (as opposed to log or another transformation), the interpretation is that there is an expected increase of $$\beta_{age}$$ units in the outcome variable for each unit change in age.
By asking whether $$\beta_{age}$$ is valid I suppose you are asking whether it is unbiased? OLS coefficients are random variables themselves, with a distribution centered around the true $$\beta$$ for the variable in question provided that Gauss-Markov assumptions are satisfied. One of these assumptions is exogeneity, which is the condition that $$Cov(independent \;variables, error)=0$$. This means that your explanatory variables cannot be correlated with any determinants of your outcome variable that are not included as explanatory variables themselves.
So your $$\beta_{age}$$ is unbiased provided that it is not correlated with any other determinants of your outcome variable that are not explicitly included in your regression model. In my field of work, this is usually an extremely heft assumption.