Modeling earthquake damage

We have data from the 2015 Gorkha earthquake in Nepal. After the earthquake, a large scale survey was conducted to determine the amount of damage the earthquake caused for homes, businesses and other structures. Variables include:

• Damage: the amount of damage suffered by the building (none, moderate, severe)
• age: the age of the building (in years)
• condition: a de-identified variable recording the condition of the land surrounding the building (n, o, or t)

We fit a multinomial regression model to predict Damage:

\[Damage_i \sim Categorical(\pi_{i(None)}, \pi_{i(Moderate)}, \pi_{i(Severe)})\] \[\log \left( \dfrac{\pi_{i(Moderate)}}{\pi_{i(None)}} \right) = \beta_{0(Moderate)} + \beta_{1(Moderate)} Age_i^{1/2} + \\ \hspace{5cm} \beta_{2(Moderate)} ConditionO_i + \beta_{3(Moderate)} ConditionT_i\] \[\log \left( \dfrac{\pi_{i(Severe)}}{\pi_{i(None)}} \right) = \beta_{0(Severe)} + \beta_{1(Severe)} Age_i^{1/2} + \\ \hspace{5cm} \beta_{2(Severe)} ConditionO_i + \beta_{3(Severe)} ConditionT_i\]

The output of the fitted model is shown below:

m1 <- multinom(Damage ~ sqrt(age) + 
                 condition, 
               data = earthquake)

summary(m1)

## Call:
## multinom(formula = Damage ~ sqrt(age) + condition, data = earthquake)
## 
## Coefficients:
##          (Intercept) sqrt(age)  conditiono conditiont
## moderate   0.6581163 0.3747641 -0.45376940 -0.5803708
## severe     0.1881145 0.4251732  0.04706934 -0.4623774
## 
## Std. Errors:
##          (Intercept)  sqrt(age) conditiono conditiont
## moderate   0.1208913 0.01684468  0.2305975  0.1155475
## severe     0.1243799 0.01725782  0.2292533  0.1180182
## 
## Residual Deviance: 18649.86 
## AIC: 18665.86