# Questions the Multiple Linear Regression Answers

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

The first category establishes a causal relationship between three or more metric variables: one continuous dependent variable and two or more independent variables. In contrast to correlation analysis, which does not indicate directionality of effects, the multiple linear regression analysis assumes that the independent variables have an effect on the dependent variable. The correlation among the variables in multiple regression analyses can be assessed with the coefficient of determination (*R ^{2}*).

Medicine: Do body weight, calorie intake, fat intake, and age have an influence on the blood cholesterol level? To answer this question the researcher would measure body weight, fat and calorie intake, as well as age, and the blood cholesterol level in various subjects. The multiple linear regression analysis can then show whether the independent variables have an effect on the blood cholesterol level (dependent variable).

Biology: Do the oxygen level, phosphor concentration, and levels of minerals in water stimulate plant growth? The research team would measure different concentrations of oxygen, nitrogen, minerals in the water used to water the plants and then measure the plants’ growth. Multiple linear regression analysis establishes whether a causal relationship between the independent and dependent variables exist. Multiple linear regression analysis is particularly useful to test observations made in experimental conditions such as this – here the levels of oxygen, nitrogen, and minerals in the water are deliberately manipulated to test their effects on growth.

Management: Do customer satisfaction, brand perception and price perception influence loyalty? The research team would ask customers to rate their satisfaction and perceptions of brand and price, as well as their loyalty to the product. The multiple linear regression analysis can then prove the assumed causal relationship of satisfaction and perceptions on loyalty.

Psychology: Is anxiety influenced by personality traits? To answer this question the team of researchers would measure anxiety (e.g. BAI) and personality trait (e.g., extroverted, introverted, etc.). Multiple linear regression analysis can be used to test whether there is a causal link between those variables. However, multiple linear regression does not prove that the causal direction is from anxiety to personality or the other way around.

Secondly, multiple linear regression can be used to forecast values:

Medicine: With X cigarettes smoked and Y hours of sport per day, the life expectancy is Y years. The research team can observe smoking and activity habits as well as age at death in a sample. The regression coefficients estimated with a multiple linear regression equation y = b0 + b1*x1 + b2*x2 can then tell the researchers by exactly what the life expectancy (y) is when smoking x cigarettes a day and working out for y hours.

Biology: By how much will 5 additional weeks of sunshine and 100mm of rain raise the sugar concentration in vine grapes? In a sample that measures the sunshine duration, the rainfall and the produced sugar level in grapes; multiple linear regression analysis can be used to establish the formula y = b0 + b1*x1 + b2*x2. Linear regression analysis is particularly useful when the x variables are not completely random.

Management: With A amount of dollars spend on brand marketing, B spend on product marketing, and C spend on in-store advertising, what is the expected sales for product Y? On a survey of different companies the researcher observe the different types of marketing spend and product sales. A multiple linear regression analysis estimates the regression function y = b0 + b1*x1 + b2*x2+ b3*x3 which can be used to predict sales values y for a given marketing spend combination A, B and C.

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

Medicine: How does the life expectancy decrease for every additional pound overweight and for every X cigarettes smoked per day? The researchers observe average daily cigarette consumptions, overweight and the age at death. Multiple linear regression analysis can be used to predict trends, e.g., for every cigarette life shortens by 2 hours; for every pound overweight life shortens by a month.

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