An article in the Journal of Pharmaceutical Sciences ( 80 , 971-977, 1991) prese

Question

An article in the Journal of Pharmaceutical Sciences (80, 971-977, 1991) presents data on the observed mole fraction solubility of a solute at a constant temperature, along with x1 = dispersion partial solubility, x2 = dipolar partial solubility, and x3 = hydrogen bonding Hansen partial solubility. The response y is the negative logarithm of the mole fraction solubility.

a) Fit a complete second order model to the data.

b) Test for the overall significance of the regression.

c) Examine the residual plots and comment on the model adequacy.

d) Report R2 and R2adj. Which gives a better assessment of the models predictive

ability?

e) Test whether the second order terms are significant to the regression.

y                x1   x2   x3
0.222         7.3   0      0
0.395         8.7   0      0.3
0.422         8.8   0.7   1
0.437         8.1   4      0.2
0.428          9       0.5   1
0.467         8.7    1.5   2.8
0.444         9.3   2.1   1
0.378         7.6   5.1   3.4
0.494          10      0      0.3
0.456         8.4   3.7   4.1
0.452         9.3   3.6   2
0.112         7.7   2.8   7.1
0.432         9.8   4.2   2
0.101         7.3   2.5   6.8
0.232         8.5   2       6.6
0.306         9.5   2.5   5
0.0923       7.4   2.8   7.8
0.116         7.8   2.8   7.7
0.0764       7.7   3        8
0.439        10.3   1.7   4.2
0.0944       7.8   3.3   8.5
0.117         7.1   3.9   6.6
0.0726       7.7   4.3   9.5
0.0412       7.4   6       10.9
0.251         7.3   2        5.2
0.00002   7.6   7.8   20.7

Explanation / Answer

a.

b.

Ho1: 1=0, 1 is not significant

v/s H11: 10, 1 is significant.

p-value = 0.000

Since p-value < 0.05, reject H01 at 5% level of significance and conclude that 1 is significant.

Ho2: 2=0, 2 is not significant

v/s H22: 20, 2 is significant.

p-value = 0.02

Since p-value < 0.05, reject H02 at 5% level of significance and conclude that 2 is significant

Ho3: 3=0, 3is not significant

v/s H13: 30, 3is significant.

p-value = 0.000

Since p-value < 0.05, reject H03at 5% level of significance and conclude that 3is significant.

Ho: model is significant

h1: model is not significant

with f=47 and p-value < 0.05, i reject ho and conclude that model is significant

c.

The first three plots shows a random pattern, indicating a good fit for a linear model.

d)

r^2 tells the variation in y which is expalined by all independent variables.

adj R^2 tells variation in y which is expalined by all significant independent variables.

SUMMARY OUTPUT Regression Statistics Multiple R 0.930984 R Square 0.866732 Adjusted R Square 0.848559 Standard Error 0.065815 Observations 26 ANOVA df SS MS F Significance F Regression 3 0.619763 0.206588 47.69348 8.54E-10 Residual 22 0.095294 0.004332 Total 25 0.715057 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept -0.36931 0.143646 -2.57095 0.01743 -0.66721 -0.0714 -0.66721 -0.0714 x1 0.08651 0.016003 5.405937 1.98E-05 0.053322 0.119698 0.053322 0.119698 x2 0.024412 0.010226 2.387334 0.025994 0.003205 0.045618 0.003205 0.045618 x3 -0.02858 0.004477 -6.38252 2.01E-06 -0.03786 -0.01929 -0.03786 -0.01929 Regression Equation: y = 0.36931 + (0.08651*x1) + (0.024412*x2) - (0.02858*x3)

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