W2 Correlations and Predictions

Question 1

What does no correlation look like?

Answer 1

Question 2

What does positive correlation look like?

Answer 2

Question 3

What does negative correlation look like?

Answer 3

Question 4

What does covariance tell us?

Answer 4

Covariance is a measure of how much two random variables vary together. The magnitude of their relationship. Their directionality ( + or -)

Question 5

What is the formula for covariance?

Answer 5

SUM[i-n] (x[i] - μ(x))(y[i] - μ(y)) / n

Question 6

What are the problems with covariance? σXY

Answer 6

The variables are centred, but not to scale. If Cov(X,Y) = 3.9 and Cov(Z,Q) = 5.2, we know both pairs are positively correlated, but we don’t know which one has the stronger correlation, because they could be in different scales/units.

Question 7

How to scale covariance with Z scores?

Answer 7

z = (x – μ) / σ Divide it by the standard deviation. Standardized scores are called z-scores. SUM[i-n] ZxZy / n #for each x and y in data.

Question 8

How to scale covariance from raw scores? ρ

Answer 8

Start with covariance. Replace (x - μ)(y - μ) with the ((x – μ) / σ) ((y – μ) / σ) Simplify and it becomes: σxy / Sqrt(σx^2 σy^2)

Question 9

How do we determine if correlation means causation?

Answer 9

Run an experiment, explicitly manipulate independent variable, one at a time

Question 10

What is the linear regression formula?

Answer 10

Y(hat) = b[0] + b[1]X

Question 11

What’s the difference between Y and Y[hat]?

Answer 11

Y is the actual real life value on plotted on the graph, Y[hat] is the predicted value

Question 12

What are Residuals?

Answer 12

Vertical deviations from a point (dot) to the line

Question 13

What is the formula for SSresidual/SSerror?

Answer 13

SUM[i-n] (Y[i] - Y[i hat])^2

Question 14

How do we calculate INTERCEPT & SLOPE from SSerror?

Answer 14

Start with formula, then sub in b[0] + b[1]X in place of Y[i.hat]. SUM[i-n] (Y[i] - Y[i hat])^2 SUM[i-n] (Y[i] - b[0] + b[1]Xi)^2 Then rearrange to make b1 or b0 the subject. b0 = Y[mean] - b1X[mean]

Question 15

What are the key assumptions of linear regression?

Answer 15

Linear relationship (straight, not curve)

Homoscedasticity (not a cone, equal distrubution)

Normality of residuals (On both extremes on ends cancel out/match)

Question 16

What does Heteroscedasticity look like vs Homoscedasticity?

Answer 16

Homo is even

Question 17

What is confidence?

Answer 17

A shaded part of around the line. How sure you are that the values fall within the shaded part. Usually uses a confidence interval of 95%.

Question 18

What is overfitting?

Answer 18

When the line fits the graph too strictly and bends for noise.

Question 19

What is multiple regression?

Answer 19

When you use many X to predict one Y

Question 20

What is the formula for multiple regression?

Answer 20

̂ Y = b0 + b1X1 + b2X2 + .. + bnXn

Question 21

What is ρxy? How do you get there?

Answer 21

A centred scaled measure for correlation.

Start with the Covariance. For the z score of x, divide the Xi - Mx by the standard deviation.

Then simplify.

Question 22

How can we calculate intercept and slope using the SSerror?

Answer 22

Question 23

What is the formula for the SSerror? (Or SSresidual)

Answer 23

It’s the sum of all squared residuals