YR1SM1: Statistics part 2

How to find an equation, represented by a regression line, that estimates linear-related data (3)

choose two points through which a line would best fit with the rest of the data - find the slope and y intercept - use for predictions

Residual =

the error, or the difference between the observed value of y (y) - the predicted value of y (ŷ) using the regression line

Least-squares regression line

a line that minimizes the sum of the squared residuals/errors

The equation of the least-squares regression line is given by…

ŷ = (b₁)(x) + b₀

bsub1 represents what in the least-squares regression line?

( r ) * (Sy / Sx) ← the slope

bsub0 represents what in the least-squares regression line?

bsub0 = ȳ - (bsub1)(x̄) (y intercept)

When interpreting bsub0 / the y-intercept of the least squares regression line, what must you note?

Y-intercept would be used to estimate the response variable at an explanatory variable value of 0, but

-if 0 is not a reasonable value for the explanatory variable

-if no observations are close to 0

THEN DO NOT interpret the y intercept

Coefficient of determination purpose and range?

measures the proportion of total variation in the response variable that is explained by the least-squares regression line; essentially measures how well of a fit your model is for the actual data

0 ≤ R² ≤ 1

R² = 0 indicates

R² = 1 indications

line has no explanatory value

the line explains 100% of the variation in the response variable

Total deviation is the difference between…

y - ȳ (observed - mean)

Explained deviation is the difference between…

ŷ - ȳ (predicted - mean)

Unexplained deviation is the difference between…

y - ŷ (observed - predicted)

Why is the difference between the predicted value and the mean value considered explained deviation?

This difference can be explained using information from the linear regression model we created: the ȳ sets the standard for the values of the y/response variable, but since we use a linear line to predict the values, of course ŷ and ȳ values won’t match up

Why is the difference between the observed value and the predicted value considered unexplained deviation?

ŷ only predicts the expected value if the explanatory variable was the only one affecting the outcome, however there’s always the chance of lurking variables or just error

Why does R² =

(Explained variation / total variation) =

1 - (unexplained variation / total variation)

R² measures how well the linear regression model fits with a maximum value of 1; explained variation can be explained by the model, so the more variation that cannot be explained, the less the model fits

What is a residual?

the difference between the predicted and observed/actual y values, aka error

Key purpose of residuals in residual analysis

determines whether a linear model is appropriate

How is a residual plot laid out?

Residuals is plotted on the y-axis and the explanatory variable is plotted on the x-axis

How does a residual plot determine how well a linear model fits?

If the points are all over the place → LINEAR MODEL GOOD

If the points follow a certain pattern → LINEAR MODEL NOT A GOOD FIT

What is constant error variance?

the error/residual values for each point stay relatively constant throughout

When is constant error variance followed and when is it violated?

FOLLOWED: the points in a residual plot have no discernable pattern about the horizontal line of residual = 0 and are ~even distances away from y = 0

NOT FOLLOWED: as the explanatory variable value grows, the points get closer/farther from residual = 0

If a residual plot shows that the points get further away from residual = 0, what does this indicate if using a linear regression model?

The linear regression model becomes less reliable as x grows bigger because there is more variance in y

What is an influential observation

an extreme-case outlier which differs greatly from the pattern of the rest of the data, thus significantly affecting either the least squares regression line’s slope or y intercept, or the value of r