Lecture 9 - Regression Depression

Question 1

Why is regression analysis used so often?

Answer 1

It models a predictor for the future, or further along the gradients measured?

Question 2

How is the independence assumption evaluated?

Answer 2

By plotting the residuals against increasing observed values (fitted)
occurs when the residuals fluctuate uniformly about 0 (no pattern)
ie. no unbalanced + or - groups

Question 3

When does a lack of independence with residuals occur?

Answer 3

When adjacent residuals tend to be similar and thus appear to be correlated = autocorrelation
ie. groupings that consistently fall below or above the line

Question 4

What is positive autocorrelation (dependence?)?

Answer 4

Positive residuals are followed by other positive individuals

Question 5

What is negative autocorrelation?

Answer 5

negative residuals are followed by other negative residuals

Question 6

What is the purpose of regression depression?

Answer 6

To examine linearity between two variables

Determine if a linear relationship exists between two variables

Question 7

If the R2 value is good, what might still lead us to be concerned with what a scatter plot depicts?

Answer 7

If the scatterplot shows a plateau

Could mean constraints on data

Question 8

Do we want to fit a curve to the data (polynomial)?

Answer 8

Not desirable to curve data and fit a polynomial because it gets complicated and is not very suitable for biology
Sometimes we might transform the data in this situation

Question 9

What is deriving R2 and the decomposition of variability?

Answer 9

a least square analysis to fit the best line that reduces the variability around the dependent response variable (y)

Question 10

What is the Y-bar?

Answer 10

Mean of all x and y values (pivots variability around dependent response?)

Question 11

What is (B) in the decomposition of variability?

Answer 11

linear distance from observed to expected = residual or error or unexplained term
(yi-y-hat-i)

Question 12

What is yi?

Answer 12

the observed value

Question 13

What is y-hat-i?

Answer 13

the expected value

Question 14

What is (C) in the decomposition of variability?

Answer 14

The Model (y-hat-i - Y-bar) and is linear distance from the expected to the mean of all x and y values (Y-bar)
=model or regression #

Question 15

What is (A) in the decomposition of variability?

Answer 15

The two components of (B) and (C)

where (A) = (B)+(C) to = the total variability

Question 16

Total variation equation = ???

Answer 16

Total variation (A) = Residual (error/unexplained/B) + Model (explained/regression#/C)

Question 17

Why do we sum the squares of the residuals?

Answer 17

To get rid of negative differences from the difference between the expected yi and the Y-bar mean of the x and y’s
Negatives would make it add up to 0 or some other weird number

Question 18

What is the equation for the Total residuals (SSt)? And how does this relate to the A B C?

Answer 18

Total SSt = Residual SSe (B) + Model SSm (C)

Question 19

What is SSe a measure of?

Answer 19

SSe is a measure of how well the regression line fits the actual data (difference between observed and expected values)

Question 20

What is SSm a measure of?

Answer 20

SSm is a measure of how different the line y-hat-i is from Y-bar (how different is the slope from 0)

Question 21

What is the equation for the Coefficient of determination R2?

Answer 21

R2 = SSm (C)/ SSt (A, total)

Question 22

What is the Model referring to in an ANOVA table?

Answer 22

The treatment/factor

Question 23

How do you determine F with the ANOVA?

Answer 23

F=MSm/MSE

or F=MSm/MSres (same thing, different label)

Question 24

What is the equation for the sum of squares Model (SSm)?

Answer 24

Sum of (Y-bar - y-hat-i)2

Question 25

What is the equation for the sum of squares Residual (SSres or SSe)?

Answer 25

Sum of (yi - y-hat-i)2

Question 26

What is the equation for the sum of squares total (SSt)?

Answer 26

Sum of (yi - Y-bar)2

Question 27

What is the degree of freedom for the Residual/Error?

Answer 27

n-2

Question 28

What is the degree of freedom for the Total?

Answer 28

n-1

Question 29

How do you determine the MS (Mean Square)?

Answer 29

The sum of squares divided by the degrees of freedom

SS/df

Question 30

How do you calculated F?

Answer 30

MSmodel/MSresidual

Question 31

What is the H(0): for the ANOVA table for simple linear regression?

Answer 31

H(0): is that there is no relationship between y and x

Cannot use x to predict y

Question 32

We can usually expect the slope and intercept to not exactly equal 0, so why do we test these hypothesis anyways?

Answer 32

We test the hypothesis anyways to see if the difference from 0 is significant

Question 33

What are the 3 null hypothesis of the Regression coefficients?

Answer 33

1) The intercept is no different from 0
intercept = constant
2) The slope is no different from 0
x = slope? (in example weight=slope)
3) There is no relationship between the response and the predictor
combination of the other tests?

Question 34

There are 2 t-tests in the regression coefficients. Why?

Answer 34

Because they test each coefficient and their hypothesis separetly

Question 35

What is done to test the assumption of linearity?

Answer 35

a scatterplot with a linear regression line and 95% confidence intervals plotted
If the data points are weakly scattered about the regression line then a linear regression may not be appropriate

Question 36

How do you fix the linearity assumption if the points are weakly scattered about the regression line?

Answer 36

Plot a curvilinear relationship (plateau problem?)

Question 37

What is done to test the assumption of normality?

Answer 37

A normal Q-Q plot is plotted and if the residuals come from a normal distribution the standard residuals should appear to fall on the line of the plot. If they skew off on the sides, that could indicate tails.
ie. the plot should track the straight line

Question 38

What is the data ordered like in a Residual Diagnostics plot (4 in 1)?

Answer 38

Ordered smallest to largest and standardized to 0 for itself and the relative variability must be the same (no patterns)