Bio-statistics linear regression

Question 1

The relationship between outcome (Y) and a covariate (X) can be

Answer 1

either linear or non‐linear

Question 2

the outcome is ……

and the exposure is ……

Answer 2

 The outcome is continuous.

 The exposure can be continuous or categorical.

Question 3

A scatter‐plot can help determine:

Answer 3

 Is the relationship between outcome & covariate linear?
 How strong is the strength of the relationship?

Question 4

correlation coefficient

Answer 4

The strength of the relationship can be negative, zero or positive,
and assessed by the correlation coefficient.

Question 5

Outcome” (Y) other names

Answer 5

 Response variable

 Dependent variable

Question 6

“Exposure” (X) other names

Answer 6

 Covariate
 Independent variable
 Predictor
 Explanatory variable
 Risk factor

Question 7

Similarities between CORRELATION AND REGRESSION

Answer 7

 Create a scatter plot of outcome vs exposure. Observe the pattern.
 Outcome is continuous
 Exposure: continuous or categorical
 Hypothesis test used for both.
 Correlation: r = 0 vs r ≠ 0.
 Regression:  = 0 vs  ≠ 0.
 AIM: To find if there is an association between the chosen exposure and outcome.

Question 8

Differences between CORRELATION AND REGRESSION

Answer 8

 Correlation:
r ranges from ‐1 to +1.
Strength of relationship.
 Regression:
 B‐coefficient can be any value.
Equation: outcome & exposure.
Predict the value of outcome
from a certain exposure value.
Two types of regression:
Simple
Multiple

Question 9

LINEAR REGRESSION – STEPS

Answer 9

1. Graph the data. Check linear relationship.
2. Calculate correlation coefficient
3. Do linear regression analysis
4. Evaluate the model
    Coefficient of determination (R2)
    Residual plot
    Normal probability plot

Question 10

CORRELATION COEFFICIENT

Answer 10

 Correlation coefficient, p, quantifies the linear relationship between a pair of variables.
 The correlation coefficient can be between ‐1 and +1.
Stats package (Graph Pad, SPSS, Stata) used to obtain “r” .
Degrees of freedom: n ‐ 2

Question 11

What is the  Hypothesis test for correlation:

Answer 11

 Null: Correlation = 0

 Alternative: Correlation ≠ 0

Question 12

HOW TO INTERPRET A CORRELATION COEFFICIENT?

Answer 12

 r < 0.00 (Negative numbers)
Negative relationship. As X increases, Y decreases.
 r > 0.00 (Positive numbers)
Positive relationship. As X increases, Y also increases.

Question 13

Ranges of r (magnitude)

Answer 13

Ranges of r (magnitude)
 0 to 0.3 = fairly weak
 0.3 to 0.7 = fairly strong
 0.7 to 0.9 = strong
 Above 0.9 = very strong

Question 14

THREE ASSUMPTIONS OF LINEAR REGRESSION

Answer 14

The outcome (Y) variable follows a normal distribution.
 Check by histogram or boxplot.
The relationship between outcome (Y) and covariate (X) is linear.
 Check with a Scatterplot.
There is constant variance of the outcome across different values of the covariate.
 Check with a residual plot

Question 15

Two types of linear regression models:

Answer 15

 Simple – one risk factor.

 Multiple – at least two risk factors

Question 16

Equation of a simple regression line (one x variable):

Answer 16

y= B0+B1X1
B1 = slope of the line.
B0 = Y‐Intercept
x1 = The value of variable “x”.

Question 17

WHAT DOES B1 REPRESENT?

Answer 17

The beta coefficient represents the amount of change in outcome variable for every unit change in the covariate, that is, the effect of the covariate on the outcome.

Question 18

t‐score follows a t‐distribution with df = n – 2

Answer 18

t=B1/SE

Question 19

The 95 % Confidence Interval

Answer 19

Statistic +,- Multiplier x Standard Error

= B1 +,- t xSE

Question 20

What are the 3 ways to evaluate linear regression model?

Answer 20

 There are 3 ways to evaluate the linear regression model:
1. Coefficient of determination (R2)
2. Residual plot
3. Normal Probability Plot
These evaluate whether there are any outlier data points.
Outliers can have a large influence on the regression equation.

Question 21

COEFFICIENT OF DETERMINATION (R‐SQUARED)

Answer 21

 The coefficient of determination tells us about the proportion
of variation in the outcome variable that is explained by the
covariate(s).
 It is the square of the correlation coefficient. i.e. R2 = r2
 R2 can range from 0 to +1. (r ranges from ‐1 to +1.)

Question 22

What is the “residual”?

And how to calculate it?

Answer 22

 The “error” between the “observed” and “predicted value”.
i.e. How far away from the “line of best fit” is the point?

Residual = Observed value – Predicted value (from equation)

Question 23

Residual plot of a linear regression

Answer 23

For linear regression:
 The residuals are random.
 They follow a normal distribution

Question 24

NORMAL PROBABILITY PLOT

Answer 24

Why is it done?
 To check if the outcome (Y) variable is normally distributed.
 If the dots follow a straight line, the data is normal.
 If the dots are scattered at either tail, the data is skewed.

Not possible in Graph Pad.
Can be done in Excel (Data Analysis – Regression)

Question 25

Stepwise regression

Answer 25

– a method of selecting significant factors in above.