Surfstat.australia: an online text in introductory Statistics
SUMMARISING AND PRESENTING DATA
PRESENTING DATA FOR TWO CONTINUOUS MEASUREMENTS
Notes
- The line y =
+ 
x is called the regression
line of y on x (the least squares regression line)
- x is called the predictor or explanatory variable
(sometimes the independent variable)
- y is called the outcome or response variable
(sometimes the dependent variable)
The idea can be generalised to equations involving more than one
predictor
- Some calculators do least squares regression - if yours does check
the moisture equation.
- The MINITAB command for y = a + bx is
More generally, e.g.
- Relationship between correlation and regression
Correlation coefficient is 
so x and y are interchangeable.
Regression line is y =
+
x where
=
-
and
=
so x and y are not interchangeable.
If you treat x as the response and y as the predictor you get a
different line because you would minimise the horizontal distances
rather than the vertical distances.
i.e. the square of the correlation coefficient r is the product of
the slopes of these two lines. If the linear relationship is perfect
the two lines will be the same: then because "X" and "Y" are
interchanged, each slope will be the reciprocal of the other,
and their product will be 1.
- Lines estimated from small data sets or widely scattered points
(i.e. with low correlation) will be subject to error and predictions
made from them may be unreliable.
- Always look at the scatter plot before fitting a regression line to
see if a linear relationship is sensible.
If the plot (or theory) suggests a non-linear relationship it may be
possible to either transform the data into a linear relationship and
then fit a straight line or else use a non-linear function of the
predictor variables.
Example - American female runners
The table below shows speeds (in feet per second) and stride rates
(number of steps per second) of some of the best American female
runners.
Moore and McCabes "Introduction to the Practice
of Statistics".
Speed (x) 15.86 16.88 17.50 18.62 19.97 21.06
Stride Rate(y) 3.05 3.12 3.17 3.25 3.36 3.46
Data were entered into a MINTAB worksheet in columns C5 (speed) and
C6 (stride).
MTB> Plot 'stride' 'speed';
SUBC> Symbol 'x'.
Analysis of Variance
SOURCE DF SS MS F p
Regression 1 0.11789 0.11789 1807.69 0.000
Error 4 0.00026 0.00007
Total 5 0.11815
MTB > let c3 = 1.79677 + 0.078527 * c5
MTB > let c4 = c6 - c3
MTB > name c3 'fit' c4 'residual'
MTB > print c5 c6 c3 c4
ROW speed stride fit residual
1 15.86 3.05 3.04221 0.0077918
2 16.88 3.12 3.12231 -0.0023060
3 17.50 3.17 3.17099 -0.0009923
4 18.62 3.25 3.25894 -0.0089428
5 19.97 3.36 3.36495 -0.0049543
6 21.06 3.46 3.45055 0.0094514
MTB > plot c4 c5
