Relating Two Variables: Linear Regression and Correlation

Question 1

What can you do first to analyse how one variable relates to another

Answer 1

A scatter plot
(However you still need a QUANTITATIVE DESCRIPTION of the plot)

Question 2

What does fitting a line to the data in a scatter plot allow for

Answer 2

Allows for a QUANTITATIVE DESCRIPTION

Question 3

What is the straight line formula

Answer 3

Question 4

What is the most widely used algorithm for finding the slope and intercept

Answer 4

METHOD OF LEAST SQUARES
Counts the distances of the points above the line as positive but as a negative for points below the line then square the distances before adding them up

Question 5

What does the fitted line in the scatter plot still require

Answer 5

Statistical Context

Question 6

What is the dependance on the mean of the Y variable on the X variable known as

Answer 6

Regression of Y on X

Question 7

What do you have to consider to analyse two variables

Answer 7

1. The MEAN of the POPULATION of ONE VARIABLE depends LINEARLY on the VALUE of the OTHER VARIABLE (so the mean will vary linearly with other variable). REMEMBER DON’T assume that a variable of the individual depends on the other viable via a straight line relationship_
2. Assume the SPREAD of the y variable about this mean is measured by STANDARD DEVIATION (alpha sign) about the line and DOESN’T CHANGE with the X variable

Question 8

What are the three parameters to estimate in a Regression Analysis

Answer 8

1. The intercept defining the mean (α)
2. The slope of the line defining the mean (β)
3. Standard Deviation about the line

Question 9

What can regression analysis sometimes introduce into the analysis

Answer 9

Asymmetry analysis

Question 10

What does the asymmetry in the regression analysis sometimes mean

Answer 10

It sometimes means regression is NOT THE RIGHT TOOL to analysis the two variables. However, this problem can be best posed in this assemetruc way

Question 11

How do you estimate the population mean by regression analysis?

Answer 11

By α+βx (population intercept +slope)
(They do this by the sample intercept+ slope)

Question 12

How do you estimate the standard deviation by regression analysis?

Answer 12

By a Quantity related to the MINIMISED SUM OF SQUARES about the fitted line

Question 13

What does a standard deviation estimate in regression analysis

Answer 13

The spread of data

Question 14

What are the units of the intercept

Answer 14

The same units as the Y variable

Question 15

What are the units of standard deviation about the line?

Answer 15

Same units as Y variables

Question 16

What are the units of the slope

Answer 16

Y per X e.g. litres per cm

Question 17

How good is the intercept (α) in estimating and testing a hypothesis

Answer 17

LITTLE DIRECT INTEREST as regression analysis is aimed at ELUCIDATING the RELATIONSHIP BETWEEN Y and X

Question 18

How good is the intercept (β) in estimating and testing a hypothesis

Answer 18

It is of GREATER INTEREST because it MEASURES the RATE AT WHICH THE MEAN OF THE Y VARIABLE CHANGES AS THE X VARIABLE CHANGES

Question 19

How do you see how good the sample slope (b) is as an estimate of β?

Answer 19

You measure STANDARD ERROR which is calculated by MINITAB

Question 20

What does β=0 mean?

Answer 20

1. The mean of Y variable DOESN’T change with the X variable - FORMS OF LINEAR DEPENDANCE
2. NO ASSOCIATION between y and x variables

Question 21

Does standard error play an important role in the testing of the hypothesis

Answer 21

YES

Question 22

What computer program fits a linear regression?

Answer 22

Minitab- it can fit much more elaborate modes
(Standard Error of β is given under the heading SE Coef and S is standard deviation)
(The test of hypothesis β=0 is based on the t-statistic given under T-value with a corresponding P-value)

Question 23

What are the Key items in minitab?

Answer 23

1. Estimate slope and intercept =given under Coef
2. Standard Error of the slope = given under SE Coef
3. P-value for the test of the hypothesis β=0
4. Standard Deviation about the line= given as S

Question 24

What is the intercept

Answer 24

It is the MEAN of Y variable when the individual has a X variable EQUAL TO 0

Question 25

Can regression be used to predict one variable from another

Answer 25

Yes regression can be used to predict the value of another variable from the measurement of one value of another

Question 26

Why is predicting the value of another variable from the regression of another variable important and what is it important for?

Answer 26

1. Predicting variables that would be difficult or invasive to measure 2. For EQUALLY VARIABLE which only become apparent in the FUTURE such as survival time can be predicted from variables known as presentation

Question 27

When making predictions from regression analysis what is important to take into account?

Answer 27

NATURAL VARIABILITY IN THE SYSTEM This often places wide limits on the predictions made for an individual More detailed consideration of preduction methods is instructive

Question 28

To predict value what estimates do you have to use

Answer 28

You have to use the estimates of β and α which would have been obtained from regression analysis

Question 29

What pairs of limits can you place on the predictions made by linear regression

Answer 29

1. 95% Confidence Intervals 2. 95% prediction intervals

Question 30

What happens to the width of the 95% confidence intervals when the sample size increases?

Answer 30

The Width of the interval decreases

Question 31

Does the bigger the sample mean a more precise estimate of the population mean (u)

Answer 31

Yes

Question 32

Can the uncertainty of the intercept and slope be quantified by the confidence intervals?

Answer 32

Yes

Question 33

When is a prediction interval contemplated and applied?

Answer 33

To an individual, prediction consisting of a single value is useful and much more helpful if an INTERVAL can be given which the VALUE OF THE INDIVIDUAL IS LIKELY TO LIE

Question 34

Is prediction intervals the same as confidence intervals

Answer 34

No

Question 35

Can prediction interval for the prediction of the value for an individual shrink to zero width under any circumstances

Answer 35

No any sensible interval; for the prediction of the value for an individual CAN'T SHRINK TO ZERO WIDTH UNDER ANY CIRCUMSTANCES

Question 36

In prediction intervals, when increasing the sample size is there change in the width of the intervals

Answer 36

There is LITTLE CHANGE IN THE PREDICTION INTERVAL but the confidence intervals become narrower

Question 37

What does little change in the prediction interval when the sample size increases and the confidence interval narrows highlight?

Answer 37

NATURAL VARIABILITY of the variable REMAINS UNCHANGED

Question 38

What are the pitfalls to regression analysis?

Answer 38

1. The technique is rather weak, empirical technique which deduces a relationship between two variables from that which can be apprehended in the sample itself. Therefore regression analysis might be guided by some sort of COMPARTMENTAL MODEL 2. SHOULDN'T BE USED IN SOME SORT OF DATA such as older males 3. Beware of OUTLYING OR UNUSUAL VALUES. This can have a noticeable influence on the estimates you obtain 4. It is up to the analyst to ensure that correct approach is chosen

Question 39

What are the assumptions of linear regression?

Answer 39

1. The MEAN of the Y VARIABLE at a given value of the X VARIABLE changes LINEARLY with X 2. The SPREAD OF DATA about this line is CONSTANT, that DOESN'T CHANGE as X CHANGES 3. The deviations from the line follow a NORMAL DISTRIBUTION but this is only needed if you intent to compute confidence or predictions or to perform hypothesis tests

Question 40

How do you assess linearity assummptions

Answer 40

Draw a scatter plot of the Y variable against the X variable and you can assess by eye

Question 41

How do you assess the spread about the line

Answer 41

From a scatter plot but defining quantities known as RESIDUALS

Question 42

What are residuals

Answer 42

The vertical distance of a point from the fitted line

Question 43

Are residuals positive above the line or negative below the line or can they be both

Answer 43

They can be both

Question 44

How can you tell if the fitted line truly reflects the data structure

Answer 44

The RESIDUALS are a sample from a distribution with POPULATION MEAN EQUAL TO ZERO and they have the SAME STANDARD DEVIATION

Question 45

How do you assess the normality of the deviation from the line?

Answer 45

To do a NORMAL PROBABILITY PLOT

Question 46

What does correlation coefficient attempt to quantify?

Answer 46

The DIFFERENCE BETWEEN TWO DATA SETS

Question 47

Does correlation go together with regression?

Answer 47

Yes they go together

Question 48

What are the properties of correlation (r)?

Answer 48

1. Always take VALUES BETWEEN -1 and 1 2. IF THE POINTS WERE TO LIE EXACTLY ON A STRAIGHT LINE THEN r WOULD EITHER BE -1 OR 1 3. A VALUE of 0 CORRESPONDS TO NO LINEAR RELATIONS BETWEEN VARIABLES 4. IT CAN BE COMPUTED FOR DATA WHICH COMPRISES OF CONTINUOUS VARIABLE PAIRS

Question 49

What does a Negative Correlation (r) look like?

Answer 49

The data which the Y variable tends to DECREASE as the X variable INCREASES

Question 50

What does a Positive Correlation (r) look like?

Answer 50

The Y and X Variable tend to increase or decrease together

Question 51

What is the problem with correlation?

Answer 51

Hard to interpret values that are not at the extreme. This is bad as in large samples, a WEAK CORRELATION can STILL HAVE SIGNIFICANCE.

Question 52

What method compares two methods that measure the same thing

Answer 52

Altman and Bland Method

Question 53

What does the Altman and Bland Method work with?

Answer 53

Works with the differences between individual observations

Question 54

What does the Altman and Bland method consider

Answer 54

The distribution of the differences between individual observations

Question 55

In the Altman and Bland method how is the dispersion of differences summarised?

Answer 55

Through the limits of agreement. 95% of differences will lie between these limits so if one observation on one method is , then the observation using method Y would generally not differ from X by more than what is quantified in the limits of agreements

Question 56

What does the Altman and Bland Method show

Answer 56

1. Shows QUANTITATIVELY the DEGREE OF AGREEMENT BETWEEN THE TWO METHODS and whether they agree but not a statistical one 2. They also can provide further information for example the standard deviations of the methods may be different maybe due to one method have a higher error

Question 57

What are residuals

Answer 57

Spread about the line measured by SD of mean Y

Question 58

What happens to the confidence limits when the sample size increases

Answer 58

Gets narrower

Question 59

What happens to the prediction limits when the sample size increases

Answer 59

Nothing, it does not vary with sample size

Question 60

Are the confidence intervals narrower compared to teh prediction limits at 95%

Answer 60

They are narrower and more "curved"

Question 61

What does correlation measure

Answer 61

The STRENGTH of a relationship

Question 62

What is the value of Pearson coefficient

Answer 62

Between -1 and +1

Question 63

What does the Pearson coefficient value of 1 represent

Answer 63

All observation lie on a STRAIGHT line with a positive gradient

Question 64

What kind of gradient is there if the correlation (r) is below 0

Answer 64

Negative gradient

Question 65

What does 0 represent in correlation

Answer 65

There is no linear relationship as the Pearson correlation assumes linearity

Question 66

Does correlation establish agreement