PROPORTIONATE HAZARD ANALYSIS

** Proportionate hazard analysis (PHA)** is a subtype of survival analysis in statistics introduced in 1972. Survival analysis is the study of observation and its subsequent event. The event can be disease, death, recovery or relapse. Earlier the scope of study was limited to event being death but now the wide scope expanded to mechanical failure of machine, crashing of stock market, onset of disease etc. In medicine survival analysis is used to study

- Leukemic patients and time in remission
- Time for developing heart disease for normal individuals
- Elderly population and time till death
- Heart transplants and time till death

Survival models can’t be done through simple multiple regression because of 2 reasons:

- Dependent variable may not be normally distributed which violates multiple regression assumptions
- Censoring issue- It occurs when we have some information about individual survival time, but we do not know the survival time exactly

PHA is one such regression model where no assumption regarding the nature of shape of distribution is made. This models assumes:

(Hazard ratio corresponding to situation. For example, equipment may fail twice at the rate per unit time as well maintained equipment. Twice indicates higher probability of failure) is a function of independent variable*Hazard rate*

The model could be represented from the following equation:

Where:

- is the resultant hazard in given values of m independent variables in given cases of x1, x2,…, xm and in respective time period.
- is baseline hazard is that hazard of individual when all independent variables are zero.

This equation is valid under 2 conditions:

- Hazard functions is not time dependent
- Log linear relationship between independent variable and hazard function

So the modified equation for considering time function is:

Here: h(t) is the function of baseline hazard, independent variable (x) and x times the log of time. 5.4 is for scaling purpose.

Practical application of PHA:

1) ** Stock market**: Survival analysis is used to analyse the stock survival time using PHA. This study was conducted for shanghai stock exchange to find out the main factors that influence performance of the quoted companies. They ignored all macro factors and concentrated just on the financial data. Stock survival time is the duration when the stock price reached its highest and dropped to 40% below of that value. They considered following 6 factors:

- Net Asset Per Share (NAPS)
- Cash Flow Per Share (CFPS)
- Growth Rate Of Operating Profit (GROP),
- Earnings Per Share (EPS),
- Return On Equity (ROE)
- Percentage Of Released Non Floating Share (RNF)

Industry was divided into 14 sectors and one such sector was considered along with a dummy variable. Above 6 factors were regressed through PHA and concluded that RNF and ROE does not contribute to stock prices whereas NAPS, CFPS and GROP have positive effect on stock survival times. It was also concluded that high liquidity and growth rate earning capacity contribute to make stoke survive longer.

2) ** Equipment failure**: PHA is used to analyse the risk of failure of machines considering the effect of human participants. The study also develop a cost benefit analysis to help in identifying optimal course of action for revenue maximization. Considering operator related factors and working age of machine hazard rate was calculated. The baseline hazard was a function of age of equipment and the independent variable included the human related functions. Expected machine uptime and probability of failure was calculated given the learning curves of operator and their reduction in human error over time (compute through historical data). Expected machine uptime along with production forecasting at various skill levels of operator give us the expected production output. This study also helps in predicting the minimum level of skills required by operator to operate profitably.

3) ** Medicine**: Digoxin is used to treat heart patients and slow down the heart rate of atrial fibrillation (heart rhythm disorder) sufferers. In 2011 the medicine was said to increase mortality rate but the reason behind it was unclear i.e. whether the medicine increased the mortality rate or the medicine was prescribed to patients which were suffering from serious ill. So a study was conducted to test the dependency of mortality on digoxin medicine. Two kinds of data were collected one with patients taking medicine and other not undertaking digoxin. This data was fit into PHA model. This study was also conducted on gender basis. From the result of regression it was concluded that the medicine was associated with rise in mortality in both men and women. This study further raise the usage of medicine.

To conclude PHA is an important tool to study the impact of independent variables on dependent variables in many areas. Earlier the model was only used in medicine but now the scope widen and research is being done to expand it horizon to new areas.

*Reference:*

1) http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3227332/

2) http://www.bsu.edu/libraries/virtualpress/mathexchange/06-01/Jiayi.pdf

4) http://eurheartj.oxfordjournals.org/content/early/2012/11/14/eurheartj.ehs348.long