Question

An agronomist wanted to investigate the factors that determine crop yield. Accordingly, she undertook an experiment in which thirty greenhouses of the same size were rented. In each, the amount of fertiliser (kg) and the amount of water (litres per week) were varied. At the end of the growing season, the amount of corn (kg) harvested was recorded. Using this data set, perform the following tasks to help the agronomist.

a) Specify a multiple population regression model to find out whether there is a direct link between crop yield and the amounts of fertilizer and water used. Assuming that neither fertilizer nor water is used excessively, do you expect the slope parameters to be negative or positive? Interpret the unadjusted and the adjusted coefficients of determination. What do they tell you about how well this multiple regression model fits the data? .Test the overall utility of this multiple regression model.

Corn	Fertilizer	Water
223	100	1000
321	200	1000
158	300	1000
187	400	1000
331	500	1000
255	100	2000
362	200	2000
216	300	2000
301	400	2000
342	500	2000
353	100	3000
328	200	3000
252	300	3000
346	400	3000
343	500	3000
220	100	4000
385	200	4000
247	300	4000
390	400	4000
303	500	4000
345	100	5000
268	200	5000
380	300	5000
415	400	5000
353	500	5000
366	100	6000
376	200	6000
323	300	6000
421	400	6000
448	500	6000

Answer 1

Answer:

Multiple Linear Regression Model:

The given data is as follows:

Corn	Fertilizer	Water
223	100	1000
321	200	1000
158	300	1000
187	400	1000
331	500	1000
255	100	2000
362	200	2000
216	300	2000
301	400	2000
342	500	2000
353	100	3000
328	200	3000
252	300	3000
346	400	3000
343	500	3000
220	100	4000
385	200	4000
247	300	4000
390	400	4000
303	500	4000
345	100	5000
268	200	5000
380	300	5000
415	400	5000
353	500	5000
366	100	6000
376	200	6000
323	300	6000
421	400	6000
448	500	6000

• Regression Analysis using MS-Excel:

1.Enter the data into Excel sheet as shown above.

2.If this is the first time you have used an Excel add-in, click the File tab, otherwise skip to step 7.

3.Click Options from the list on the left.

4.Select Add-ins in the Excel Options box.

5.In the Add-in list box, select Analysis Toolbox-VBA from the Inactive Application Add-ins list.

6.Click OK.

7.Then select Data/ Data Analysis tab from the menu bar.

8.The Data Analysis dialog box will appear on the screen.

9.From the Data Analysis dialog box, select Regression and click OK.

10.The Regression dialog box will appear on the screen.

11.Place independent variable (Fertilizer and Water) in Input X Range and place dependent variable (Corn) in Input Y Range.

12.Place appropriate confidence level in Confidence Level box. (If necessary)

13. Give Output Range.

14.Click OK.

Then the MS-Excel gives the following output:

Regression Statistics
Multiple R	0.644717
R Square	0.41566
Adjusted R Square	0.372376
Standard Error	57.29273
Observations	30

ANOVA
	df	SS	MS	F	Significance F
Regression	2	63042.85	31521.42	9.602996	0.000708
Residual	27	88626.35	3282.457
Total	29	151669.2

	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	194.84	32.57803	5.980718	2.23E-06	127.9954	261.6846
Fertilizer	0.122667	0.073965	1.658451	0.108801	-0.0291	0.274429
Water	0.024846	0.006125	4.056542	0.000381	0.012279	0.037413

From the above Excel output, the multiple linear regression model is,

• Interpretation of the unadjusted and the adjusted coefficients of determination:

Unadjusted Coefficient of Determination:

From the above output, the unadjusted coefficient of determination, R² = 0.4157

The coefficient of determination is a measure used in statistical analysis that assesses how well a model explains and predicts future outcomes. It is indicative of the level of explained variability in the data set. More commonly, it is used as a guideline to measure the accuracy of the model. Here, the coefficient of determination is 0.4157. Thus we can interpret that, approximately 41.57% of the observed variation can be explained by the model.

Adjusted Coefficient of Determination:

From the above output, the unadjusted coefficient of determination, Adjusted R² = 0.3724

The adjusted R-squared is a modified version of R-squared for the number of predictors in a model. Adjusted R-squared compares the correlation of the investment to several measured models.  It explains the percentage of variation of the independent variables that affect the dependent variables. If the adjusted coefficient of determination is closer to 1, it indicates that the estimated equation of regression fits the data. Here the adjusted coefficient of determination is 0.3724. Thus, we can interpret that 37.24% of the variation in model is explained by the independent variables (Fertilizers and Water).

• Goodness of Fit:

The closer the value of R² is to 1, the better the fit, or relationship, between the two factors.

Since, R² = .4157, which indicates the fitting of the multiple linear regression model is not good.

• Test for overall utility of the multiple regression model:

The test for significance of regression in the case of multiple linear regression analysis is carried out using the analysis of variance. The test is used to check if a linear statistical relationship exists between the response variable and at least one of the predictor variables. The statements for the hypotheses are:

H₀: β₁ = β₂ = 0
H₁: β_j ≠ 0, for at least one value of j; j = 1, 2.

From the above Excel output, the test statistic F = 9.60299

At 5% level of significance, the critical F value, F_{0.05, 2, 27} = 0.000708

Decision Rule: Reject the null hypothesis, if calculated F value > Critical F value.

Conclusion:

Here, Calculated F value > Critical F value (i.e., 9.603 > 0.0007). Therefore, we reject the null hypothesis at 5% level of significance.  It is concluded that at least one coefficient out of  β₁ and  β₂ is significant. In other words, it is concluded that a regression model exists between Corn and either one or both of the independent variables (i.e. Fertilizers and Water).