Source of Variation Regression SSR(X) SSRXX SSRXXX Frrr Total of 3 1 1 1 Sum of Squares 107.9 90.2 15.3 2.4 Mean Square 26.975 169 124.8 0.845 24 =b+bX+bX+bX The objective is to find the appropriate model to estimate the population linear regression model. In order to do the analysis, consider the following: 2.1. Find all relevant extra sums of squares SSE(X) SSEXX SSEXXX 2.2. Find the coefficient of multiple determination and explain what it measures 2.3. Find all the coefficients of partial determination and explain each of them 2.4. Let us assume that the above scenario is based on the following: X: Number of Beds (NoB) X2: Income per bed occupied in R1000.00 (IpB) X3: Salaries of staff in R10 000.00 (Sal) P: Income of hospital in R1000 000.00 The company that owns 400 hospitals provided the data. The table contains an extract thereof. Y: 14.5 12 12.5 10.5 X1: 200 150 400 350 X2: R1000 1.5 1.8 0.9 0.85 X3: R10000 11 12 10 A regression model was found to fit the data and the least square method provided the following fitted response function: Y = 15 + 0.12X1 + 1.45X2 – 1.2X3 2.4.1. What is the income of the hospital if X1 = 400, X2 = 0.9, X3 = 12? 2.4.2. Interpret the coefficient of b. 2.4.3. Explain the following: = 0.0932
The Correct Answer and Explanation is:
1 Extra Sums of Squares (SSE)
We calculate each SSE by subtracting the explained variation (SSR) from the total variation (SST):
- SSE(X) = SST − SSR(X) = 107.9 − 90.2 = 17.7
- SSE(XX) = SST − SSR(XX) = 107.9 − 105.5 = 2.4
- SSE(XXX) = SST − SSR(XXX) = 107.9 − 107.9 = 0
2.2 Coefficient of Multiple Determination (R²)
R2=SSRSST=107.9107.9=1.0R^2 = \frac{SSR}{SST} = \frac{107.9}{107.9} = 1.0R2=SSTSSR=107.9107.9=1.0
Interpretation:
The coefficient of multiple determination R2R^2R2 shows how much of the variation in the dependent variable YYY is explained by the regression model. An R2R^2R2 of 1 means the model explains 100% of the variation in hospital income. This indicates a perfect fit for the data.
2.3 Coefficients of Partial Determination
These measure the additional contribution of each variable after the others are included in the model.
- RX2∣X12=SSR(XX)−SSR(X)SSE(X)=105.5−90.217.7=15.317.7=0.8644R^2_{X2|X1} = \frac{SSR(XX) – SSR(X)}{SSE(X)} = \frac{105.5 – 90.2}{17.7} = \frac{15.3}{17.7} = 0.8644RX2∣X12=SSE(X)SSR(XX)−SSR(X)=17.7105.5−90.2=17.715.3=0.8644
- RX3∣X1,X22=SSR(XXX)−SSR(XX)SSE(XX)=107.9−105.52.4=2.42.4=1.0R^2_{X3|X1,X2} = \frac{SSR(XXX) – SSR(XX)}{SSE(XX)} = \frac{107.9 – 105.5}{2.4} = \frac{2.4}{2.4} = 1.0RX3∣X1,X22=SSE(XX)SSR(XXX)−SSR(XX)=2.4107.9−105.5=2.42.4=1.0
Interpretation:
RX2∣X12R^2_{X2|X1}RX2∣X12 shows that once the number of beds is accounted for, income per bed explains 86.44% of the remaining variation.
RX3∣X1,X22R^2_{X3|X1,X2}RX3∣X1,X22 shows that staff salaries explain 100% of the remaining variation after accounting for the first two variables.
2.4 Income Prediction
Given:
- X1=400X1 = 400X1=400
- X2=0.9X2 = 0.9X2=0.9
- X3=12X3 = 12X3=12
Model:Y=15+0.12X1+1.45X2−1.2X3Y = 15 + 0.12X1 + 1.45X2 – 1.2X3Y=15+0.12X1+1.45X2−1.2X3Y=15+(0.12)(400)+(1.45)(0.9)−(1.2)(12)Y = 15 + (0.12)(400) + (1.45)(0.9) – (1.2)(12)Y=15+(0.12)(400)+(1.45)(0.9)−(1.2)(12)Y=15+48+1.305−14.4=49.905Y = 15 + 48 + 1.305 – 14.4 = 49.905Y=15+48+1.305−14.4=49.905
Predicted income: R49.91 million
2.4.2 Interpret the Coefficient of b
If b=0.12b = 0.12b=0.12, it means that for every additional hospital bed, holding other factors constant, the hospital’s income increases by R120 000.00. This shows the economic value of expanding bed capacity.
2.4.3 Interpretation of R2=0.0932R^2 = 0.0932R2=0.0932
This means that only 9.32% of the variation in income is explained by the model. This is a low value and implies that the model may not be very useful for prediction or that important variables might be missing. Such a result suggests reevaluating variable selection or model assumptions.
Conclusion (Summary)
This regression analysis explores hospital income based on number of beds, income per bed, and staff salaries. Using sums of squares, the extra contribution of each variable is measured. The R² value indicates how well the model explains variations in the data. Partial coefficients reveal the incremental value each variable adds to the model. Based on this, the model with all variables included explains the income perfectly in the example, but in practice, models may yield lower R² values like 0.0932, which highlight poor explanatory power. Model adequacy must always be evaluated in the context of the data, domain knowledge, and predictive performance.
