Question

ID      Year    CornYield       SoyBeanYield
1       1957    48.3    23.2
2       1958    52.8    24.2
3       1959    53.1    23.5
4       1960    54.7    23.5
5       1961    62.4    25.1
6       1962    64.7    24.2
7       1963    67.9    24.4
8       1964    62.9    22.8
9       1965    74.1    24.5
10      1966    73.1    25.4
11      1967    80.1    24.5
12      1968    79.5    26.7
13      1969    85.9    27.4
14      1970    72.4    26.7
15      1971    88.1    27.5
16      1972    97      27.8
17      1973    91.3    27.8
18      1974    71.9    23.7
19      1975    86.4    28.9
20      1976    88      26.1
21      1977    90.8    30.6
22      1978    101     29.4

23      1979    109.5   32.1
24      1980    91      26.5
25      1981    108.9   30.1
26      1982    113.2   31.5

27      1983    81.1    26.2
28      1984    106.7   28.1
29      1985    118     34.1
30      1986    119.4   33.3
31      1987    119.8   33.9
32      1988    84.6    27.0
33      1989    116.3   32.3
34      1990    118.5   34.1
35      1991    108.6   34.2
36      1992    131.5   37.6
37      1993    100.7   32.6
38      1994    138.6   41.4
39      1995    113.5   35.3
40      1996    127.1   37.6
41      1997    126.7   38.9
42      1998    134.4   38.9
43      1999    133.8   36.6
44      2000    136.9   38.1
45      2001    138.2   39.6
46      2002    129.3   38.0
47      2003    142.2   33.9
48      2004    160.3   42.2
49      2005    147.9   43.1
50      2006    149.1   42.9
51      2007    150.7   41.7

Use both predictors. From the previous two exercises, we conclude that year and soybean may be useful together in a model for predicting corn yield. Run this multiple regression.

a)       Explain the results of the ANOVA F test. Give the null and alternate hypothesis, test statistic with degrees of freedom, and p-value. What do you conclude?

b)      What percent of the variation in corn yield in explained by these two variables? Compare it with the percent explained in the previous simple linear regression models.

c)       State the regression model. Why do the coefficients for year and soybean differ from those in the previous exercises?

d)      Summarize the significance test results for the regression coefficients for year and soybean yield.

e)      Give a 95% confidence interval for each of these coefficients.

f)        Plot the residual versus year and soybean yield. What do you conclude?

Answer 1

Note that the variable year needs to be scaled properly before being used for regression as its scale is very large compared to the other variables.

Let's assume that time t=0 corresponds to the year 1957, t=1 corresponds to year 1958 and so on. Then use Excel Data Analysis to run the multiple linear regression with 'Corn Yield' as the 'Y-Range' and 'Soyabean Yield' and 'Time' as X - Range.

You will have following output -

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.969258235
R Square	0.939461527
Adjusted R Square	0.93693909
Standard Error	7.471509358
Observations	51
ANOVA
	df	SS	MS	F	Significance F
Regression	2	41582.00842	20791.00421	372.4421086	5.87158E-30
Residual	48	2679.5257	55.82345209
Total	50	44261.53412
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	-1.604119694	9.859131867	-0.16270395	0.871434589	-21.42723291	18.21899352	-21.42723291	18.21899352
SoyBeanYield	2.557739532	0.453729415	5.637147267	8.9397E-07	1.645455399	3.470023664	1.645455399	3.470023664
Time	0.9190585	0.187647498	4.897792453	1.14082E-05	0.541767918	1.296349082	0.541767918	1.296349082

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a)

If the multiple linear regression model is : Y = + (SoyBean Yield) + (Time), then -

Null Hypothesis - H₀:

Alternate Hypothesis - At least one of the   for i = 1,2 is not =0.

The test statistic = F = 372.4421086

Degree of freedom of numerator and denominator are 2 and 48 respectively.

p-value is less than 0.0001.

As p-value is less than significance level of 0.05, so we reject the null hypothesis and conclude that the regression model is significant. That means at least one of the coefficient is significant.

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b)

The variation in dependent variable as predicted by the independent variables is given by the coefficient of variation i.e. R-squared value.

From the output, we can see that R-Squared value is = 0.9395

So, about 93.95% of the variation in corn yield is explained by those two variabes.

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c)

The regression model is -

Corn Yield = -1.6041 + 2.5577(SoyBean Yield) + 0.9191(Time)

The coefficient are different here because of the interaction of the two predictors. Earlier you used the two variables separately to get two different models so there was no interaction of predictors. But in this case there is interaction of predictors. That causes change in coefficient value.

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d)

As the p-value for both the coefficients of 'SoyBean' and 'Time' are much smaller than the significance level of 0.05, so in both cases we would reject the null hypothesis and conclude that the coefficient values are statistically significant. Hence the two predictors are significant.

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e)

The result gives 95% confidence interval for each coefficients as -

For SoyBean - (1.6455, 3.470)

For Time - (0.5418, 1.2963)

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f)

Following are the plots -

Note that the residuals are all within a band and do not follow a particular pattern. So, the assumption of normality of residuals is valid for both the variables. Hence the model is good to be used.

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Note that I have scaled the year data to time with reference to year 1957. If you need the coefficients for the year itself, you don't need to scale it. You can directly use the year values.