Regression + Correlation

Question 1

Regression

Answer 1

Describes a relationship which isn’t deterministic between two variables one which is continuous. Allows easy visual analysis of linear /non-linear relationships - dependent and independent variables

Question 2

Regression line

Answer 2

y = a + bx ( y changes with x)

Straight line of best fit where a = y-intercept and b = slope of the line. This dependence of the mean of the y variable on the x variable is known as the
regression of y on x.

Question 3

Easiest way to assess trend between variables

Answer 3

Scatter plot

Question 4

Sum of all squares

Answer 4

Estimate a line - a line is then drawn up/down from the line to each induvidual point. This difference is squared to remove the -ve and added. The smallest number = line of best fit

Question 5

Correlation

Answer 5

test of the relationship between to variables

r = 0 is a linear straight line

Question 6

Correlation coefficient

Answer 6

r. Can vary from 1 - -1 these begins the two extremes of correlation
1 = increase in one variable leads to a linear increase in the other variable
-1 = increased in one variable leads to an linear decrease in the other variable

Question 7

Assumptions for regression analysis

Answer 7

The sample is representative of the population for the inference prediction.

The error is a random variable with a mean of zero conditional on the explanatory variables.

The independent variables are measured with no error. (Note: If this is not so, modeling may be done instead using errors-in-variables model techniques).

The independent variables (predictors) are linearly independent, i.e. it is not possible to express any predictor as a linear combination of the others.

The errors are uncorrelated

We also assume that the spread of FEV1 about this mean is measured by a standard deviation, σ, about the line and that this does not change with height.

The variance of the error is constant across observations

Question 8

Regression of y on x

Answer 8

This dependence of the mean of the y variable on the x variable μ )( =α + βxx .

Question 9

What does B measure

Answer 9

Measures the rate at which the mean of the y variable changes as the x variable changes.

Question 10

B = 0

Answer 10

Mean of the y variable does not change with the x variable. Hence no association between the y and x variables.

Question 11

Outcomes to extract from regression analysis

Answer 11

• the estimated slope and intercept, given under Coef;
• the standard error of the slope, given under SE Coef;
• the P-value for the test of the hypothesis β=0;
• the standard deviation about the line, given as S.

Question 12

a = y intercept

Answer 12

Mean value when x = 0, hence may be negative, needed for correct orientation and degree of slope

Question 13

Making predictions from regression

Answer 13

Natural variability needs to be taken into account hence wide wide limits on the prediction made for an individual are often in place

estimate for height h is α + β h,

Question 14

Intervals for prediction

Answer 14

Confidence interval - calculate the uncertainty within the sample means to consturct an interval where we are 95% sure u (population mean) will lie

NB as sample increased interval will reduce

Question 15

Prediction interval

Answer 15

Can expect to see the next data point sampled. Collect a sample of data and calculate a prediction interval. Then sample one more value from the population. If you do this many times, you’d expect that next value to lie within that prediction interval in 95% of the samples.

Crucially the prediction interval tells you about the distribution of values, not the uncertainty in determining the population mean. Account for uncertainty within the mean and data scatter hence are wider than confidence intervals

Question 16

Confidence interval

Answer 16

How well you have determined the mean. If you do this many times, and calculate a confidence interval of the mean from each sample, you’d expect about 95 % of those intervals to include the true value of the population mean.

Crucially tells you about the likely location of the true population parameter.

Question 17

Pitfalls

Answer 17

Beware of outliers - noticeable influence
Regression of x on y and y on x are very different
Not applicable to different populations

Question 18

Assumptions in regression

Answer 18

That the mean of the y variable at a given value of the x variable changes linearly with x

Spread of data = constant

Deviation from the line follows a normal distribution

Question 19

Assessing linearity

Answer 19

Draw a scatterplot exclude outliers check assumption plausible

Question 20

Assessing spread

Answer 20

Use residuals best seen in a scatterplot. Vertical distance from the line to the points.

If the fitted line truly reflects the structure of the data then the residuals are a sample from a distribution with population mean equal to zero and they all have the same SD

Question 21

Assessing the Normality of the deviations from the line

Answer 21

Residuals are deviations from the line then this assessment amounts to checking that the
residuals come from a common Normal distribution. Use a common probability plot

Question 22

Properties of r

Answer 22

1. it always takes values between -1 and 1;
2. if the points were to lie exactly on a straight line then r would be either -1 or 1;
3. a value of 0 corresponds to no linear relation between the variables;
4. it can be computed for data which comprise pairs of continuous variables.

Question 23

Method comparison

Answer 23

Comparing two methods which both measure the same thing ie manual and automated BP

Null hypothesis = no difference, scatter plot to assess visually

Question 24

1st method to compare methods

Answer 24

Analysis via paired t-test to compute the differences between each measurement for A and B

Question 25

2nd method (Bland and Altman plot)

Answer 25

Plot differences vs mean (of each pair ie A+B/2)

Trend in graph indicated one method = higher SD