# Questions the Linear Regression Answers

There are 3 major areas of questions that the regression analysis answers – (1) causal analysis, (2) forecasting an effect, (3) trend forecasting.

The first category establishes a causal relationship between two variables, where the dependent variable is continuous and the predictors are either categorical (dummy coded), dichotomous, or continuous..  In contrast to correlation analysis which does not indicate directionality of effects, the regression analysis assumes that the independent variable has an effect on the dependent variable.

Examples:

Medicine: Has the body weight an influence on the blood cholesterol level? To answer this question the researcher would measure body weight and blood cholesterol level in various subjects. The linear regression analysis can then show whether the body weight (independent variable) has an effect on the blood cholesterol level (dependent variable).

Biology: Does the oxygen level in water stimulate plant growth?  The research team would measure different concentrations of oxygen in the water and measure the growth of plants.  Linear regression analysis can then be used to establish whether a causal relationship between the independent and dependent variable exist.  It is particularly useful to test observations made in experimental conditions such as this – here the oxygen level in the water might be deliberately manipulated to test the effects.

Management: Does customer satisfaction influence loyalty?  The research team would ask customers to rate their satisfaction and also their loyalty to the company.  The analysis can then prove the assumed causal relationship of satisfaction on loyalty.

Psychology: Is anxiety influenced by personality traits?  To answer this question the team of researchers would measure anxiety (e.g. BAI) and one personality trait (e.g., consciousness).  Linear regression analysis would then be used to test whether there is a causal link between both variables.  However, it does not prove that the causal direction is from anxiety to personality or the other way around.

Secondly, it can be used to forecast values:

Medicine: With X cigarettes smoked per day, the life expectancy is Y years.  The research team can observe smoking habits and age at death of a couple of participants.  The regression coefficient estimated with a linear regression equation y = a + b*x can then tell the researchers b the life expectancy (y) is when smoking x cigarettes a day.

Biology: Five additional weeks of sunshine the sugar concentration in vine grapes will rise by X %. In a sample that measures the sunshine duration and the produced sugar level in grapes; linear regression analysis can be used to establish the formula y = a + b*x.  It is particularly useful when the x variables are not completely random.

Management: With X amount of dollars spend on marketing, sales should be Y.  On a survey of different companies the researcher observe the marketing spend and sales.  A linear regression analysis estimates the regression function y = a + b*x which can be used to predict sales values y for a given marketing spend x.

Thirdly, linear regression analysis can be used to predict trends in data:

Medicine: By how many years does the life expectancy decrease for every additional pound overweight?  The researchers observed overweight and the age at death, linear regression analysis can be used to predict trends.  This is especially useful when the regression analysis finds no significant intercept.  Then the regression coefficient can at least predict a trend (if the coefficient is significant).

Biology: With every additional week of sunshine the sugar concentration in vine grapes will rise by Y%.  Similarly to the question above, the regression coefficient indicates the general trend in data.  That is for every additional unit of x the dependent variable will increase by b*x.  This is particularly useful in experimental settings where the values of x have not been chosen at random, which violates the assumptions ANOVAs and correlations make.

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Assumptions of Linear Regression