Regression Analysis: Height versus Mother Height



Reading Minitab output for a simple linear regression model

Regression Analysis: Height versus Mother Height

The regression equation is the estimated regression

Height = 24.7 + 0.640 Mother Height equation: y = b0 + b1x

(dependent (intercept, (slope, (independent

variable, y) b0 ) b1) variable, x)

(test

(estimates) (sd of ests.) statistics) (p-values)

Predictor Coef SE Coef T P

Constant 24.690 8.978 2.75 0.009 tests H0: β0 = 0, vs. two-tailed alternative

Mother H 0.6405 0.1394 4.59 0.000 tests H0: β1 = 0, vs. two-tailed alternative

intercept, b0 (the latter is equivalent to testing for

slope, b1 linear correlation between x and y)

S = 2.973 R-Sq = 35.7% R-Sq(adj) = 34.0%

(standard error (coefficient of linear (adjusted r2, used for multiple regression)

of estimate, se) determination, r2)

(the coefficient of linear

correlation is the square root for a simple linear regression

of r2, with the same sign as the model, these tests are equivalent

slope, b1)

Analysis of Variance

(test stat.) (p-value)

Source DF SS MS F P

Regression 1 186.54 186.54 21.11 0.000 tests for overall model fit

Residual Error 38 335.76 8.84 (used for multiple regression)

Total 39 522.30

(total df = n-1 “explained variation”

for simple linear “unexplained variation”

regression) “total variation”

Unusual Observations

(ŷ: predicted value of y (y – ŷ)

(observed x) (observed y) for the observed x) sŷ (y – ŷ) se

Obs Mother H Height Fit SE Fit Residual St Resid

8 63.0 71.000 65.039 0.505 5.961 2.03R

21 63.0 58.600 65.039 0.505 -6.439 -2.20R

30 75.0 71.100 72.724 1.560 -1.624 -0.64 X

R denotes an observation with a large standardized residual

(Standardized residuals are residuals divided by se; they have a mean of 0 and a standard deviation of 1. This simplifies determining how far an observed value is from the least-squares regression line, relative to the dispersion of observations around the line. Observations with standardized residuals greater than +2 or less than –2 are considered “unusual” by Minitab.)

X denotes an observation whose X value gives it large influence.

(Observations with x values far from the mean of x have a greater impact on model parameters – such observations are said to have

high leverage. Removing such observations (which may not be warranted!) would produce substantially different parameter estimates of the slope and intercept.)

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