Supplementary Exercise 2.50 of IPS7e
------------------------------------

Measurements of the concentration of fenthion (a pesticide) in olive
oil after 28-365 days (presumably after exposure). On each measurement
day, 5 concentration values were obtained; it is not clear whether these
are multiple measurements of the same sample or different samples, or
whether the samples were measured repeatedly over time. The simple exponential
decay model would correspond to a linear relation for logarithmic 
concentrations:
  ln(conc) = ln(c0)-k*t,
and the interest is in fitting a linear regression model to logarithmic
concentrations as a function of time. The model assumes
  ln(conc_i) = beta_0 + beta_1*day_i + epsilon_i, i=1,...,25
where the errors (epsilon_i) are i.i.d. from N(0,sigma). Note that the
replication of observations for 5 different days has no direct impact on
the regression model and its assumptions.


(a)+(b)  Minitab commands and output:

MTB > WOpen "H:\VHM\VHM801\Datasets\Minitab\Chapter 2\ex02_050.mtw".
Retrieving worksheet from file: 'H:\VHM\VHM801\Datasets\Minitab\Chapter
2\ex02_050.mtw'
Worksheet was saved on 15/11/2014

MTB > Name C3 'lnconc'
MTB > Let 'lnconc' = ln('fenthion')

MTB > Fitline 'lnconc' 'days';
SUBC>   GFourpack;
SUBC>   RType 2;
SUBC>   Confidence 95.0.
Regression Analysis: lnconc versus days 

The regression equation is
lnconc = - 0.03412 - 0.000507 days

S = 0.0243845   R-Sq = 87.6%   R-Sq(adj) = 87.0%

Analysis of Variance

Source      DF        SS         MS       F      P
Regression   1  0.096286  0.0962858  161.93  0.000
Error       23  0.013676  0.0005946
Total       24  0.109962
 
Fitted Line: lnconc versus days 
Residual Plots for lnconc 

MTB > Fitline 'lnconc' 'days';
SUBC>   Poly 2;
SUBC>   GFourpack;
SUBC>   RType 2;
SUBC>   Confidence 95.0.
Polynomial Regression Analysis: lnconc versus days 

The regression equation is
lnconc = - 0.01769 - 0.000786 days + 0.000001 days^2

S = 0.0233227   R-Sq = 89.1%   R-Sq(adj) = 88.1%

Analysis of Variance

Source      DF        SS         MS      F      P
Regression   2  0.097995  0.0489974  90.08  0.000
Error       22  0.011967  0.0005439
Total       24  0.109962

Sequential Analysis of Variance

Source     DF         SS       F      P
Linear      1  0.0962858  161.93  0.000
Quadratic   1  0.0017090    3.14  0.090
 
Fitted Line: lnconc versus days 
Residual Plots for lnconc 


Answers to questions:

(a)  The fitted line plot shows a fairly equal scatter of points around
the estimated line for each of the 5 time points. Actually the line has
two points above and two points below it for all days except day 183; 
this may seem curious with 5 observations per time point but there one 
set of tied values for each time point. It is not obvious from the graph
whether the fit can be improved by allowing for a non-linear relation;
we will explore this question below in (b). Our first assessment,
however, is that the data are described reasonably well by the linear
relation.

(b)  The estimated linear relation is:  lnconc = -0.03412 - 0.000507 days.
The estimated slope gives our value for k, except for a sign change:
k=0.000507. The value of k is numerically very small but this reflects
that the change in concentration is also quite small over a large storage
time. The estimated intercept can be exponentiated to give an estimate
of the constant C0: C0= exp(-0.03412) = 0.966.

One simple way of exploring whether the fit of the model can be improved 
by allowing for a non-linear relation with time is to add a quadratic
term (of time) to the equation:
  ln(conc_i) = beta_0 + beta_1*day_i + beta_2*(day_i^2) + epsilon_i.
The fit of this model is also shown above, and visually there seems to
be some improvement by allowing a curvature (which turns out to be
positive, corresponding to an upwards bending curve). The Minitab
listing also gives a significance test for this added quadratic term:
  F=3.14, P=0.092.
Without going into details with how this is computed (this is discussed
in the VHM 802 and VHM 812 courses), we can conclude, by the non-significant
test result, that there is no clear evidence of a non-linearity in the relation.