biostats lec lesson 8 linear regression

Linear regression

strives to show the relationship between two variables by applying a linear equation to observed data.

Linear regression

One variable is supposed to be an independent variable, and the other is to be a dependent variable.

regression analysis

the relationship between X and Y is described by means of an equation of the curve which best fits the data.

regression equation

After the _________________ has been derived, values of independent variables can be substituted in order to determine the predicted values of the dependent variable.

EFFECT OF EACH INDEPENDENT VARIABLE

measured by the magnitude and sign of the corresponding regression coefficient.

LINEAR REGRESSION ANALYSIS

Applied when the relationship between X and Y can be described by a straight line

MULTIPLE REGRESSION ANALYSIS.

Applied when the effects of two or more independent variables are simultaneously considered

normality, homoscedasticity and independence.

The valid application of simple linear regression analysis entails three basic assumptions regarding the dependent variable, Y, namely ________________.

method of least-squares

The most common method for fitting a regression line is the method of least-squares.

method of least-squares

This method calculates the best-fitting line for the observed data by minimizing the sum of the squares of the vertical deviations from each data point to the line (if a point lies on the fitted line exactly, then its vertical deviation is 0).

first step

finding the equation of the line which best fits the data.

proximity

However, if one were also to consider the broken lines besides C, choosing between the two lines will now be a difficult decision, unless a more specific definition of "____________" will be given.

proximity

the concept of "________________" to the given data is also used in determining the best fitting line, but it is expressed in measureable terms by the method of least squares.

X

is the independent variable and plotted along the x-axis

Y

is the dependent variable and plotted along the y-axis

b

The slope of the line is __________

a

is the intercept (the value of y when x = 0).

method of least squares

Using the _______________ as criterion for selecting the best-fitting line, the values of the slope and the intercept should be computed using the formulas.

corollary

A _________________ question that may be asked is the extent to which the independent variable, X, can be used to predict values of the dependent variable, Y.

R2

The first approach is to determine the value of the coefficient of determination.

β

The second is to test the null hypothesis that the regression coefficient for the population, __________ , is equal to zero.

THE COEFFICIENT OF DETERMINATION, R2

measures the proportion of the total variability in the dependent variable, Y, that can be explained by or attributed to the independent variable, X. It can be computed by getting the square of the correlation coefficient, r.

1. For any fixed value of X, Y has a normal distribution. 2. The variance of Y is the same for any value of X.

3. The value of Y at one value of X does not depend on, and is not affected by the value of Y at another value of X.

These are the assumptions: