Comparing Two Regression Tests
Sum of Squares | df | Mean Square | F | P-value | ||
Regression | 41.0379 | 1 | 41.0379 | 72.2735 | .000 | |
Residual | 9.6528 | 17 | 0.5678 | |||
Total | 50.6907 | 18 |
The ANOVA table above shows that there is a strong evidence to reject the null hypothesis as the p value is less than 0.05. Therefore, we conclude that there is a linear relationship between the female mass and egg number for species 1.
coefficient | Std. Error | t | P-value | ||
(Constant) | 4.1292 | 1.4409 | 2.8656 | 0.0107 | |
Female Mass | 0.2011 | 0.0237 | 8.5014 | .0000 |
Null hypothesis: there is relationship between female mass and egg number in species2
Alternative hypothesis: there is no relationship between female mass and egg number in species2
Linear regression test
Sum of Squares | df | Mean Square | F | P-value | ||
Regression | 105.1745 | 1 | 105.1745 | 154.1147 | .000 | |
Residual | 30.7099 | 45 | 0.6824 | |||
Total | 135.8844 | 46 |
Coefficient | Std. Error | t | P-value | ||
(Constant) | 4.9654 | 0.9759 | 5.0880 | 0.000 | |
Female Mass | 0.1875 | 0.0151 | 12.4143 | 0.000 |
Independent T-test
This is used to test the predicted values and species (grouping variable i.e. 1 and 2). The result is displayed below:
Species | N | Mean | Std. Deviation | Std. Error Mean |
1 | 19 | 61.0065954 | 6.94321843 | 1.59288355 |
2 | 47 | 63.8909508 | 7.11113481 | 1.03726562 |
F | t | df | Sig. (2-tailed) | Mean Difference | |
.280 | -1.502 | 64 | .138 | -2.88435533 |
The test shows that there is no sufficient evidence to reject the null hypothesis as the p-values is greater than 0.05, therefore, we conclude that the true difference between the group mean is zero i.e. the mathematical relationship between female mass and egg number is the same in both species.
Nested ANOVA Test
Null hypothesis: Polycarp General Hospital (Smyrna) requires few number of days post op before release than Saint Genesius General Hospital (Rome)
Alternative hypothesis: Saint Genesius General Hospital (Rome) requires few number of days post op before release than Polycarp General Hospital (Smyrna)
The appropriate test for this question is the Nested ANOVA test, because we need to compare the means of both hospital to determine the better choice based on getting patient home sooner.
Nested ANOVA | ||
Symra G.H | Rome G.H | |
Mean | 30.97143 | 28.37143 |
Variance | 4.057143 | 4.584679 |
Observations | 70 | 70 |
df | 69 | 69 |
F | 0.884935 | |
P(F<=f) one-tail | 0.306529 | |
F Critical one-tail | 0.671141 |
Tests of Normality | ||||||
Kolmogorov-Smirnov | Shapiro-Wilk | |||||
Statistic | df | Sig. | Statistic | df | Sig. | |
Smyra G.H | .158 | 70 | .000 | .964 | 70 | .042 |
The Kolmogorov-Smirnov test shows that the test is significant, therefore the data for Symra General Hospital is statistically significance, i.e. normality assumption test is valid.
Normality test for Rome General Hospital
Linear Regression Test
Sum of Squares | df | Mean Square | F | P-value | ||
Regression | 1112490.230 | 1 | 1112490.23 | 14.766 | .002 | |
Residual | 1130146.277 | 15 | 75343.085 | |||
Total | 2242636.507 | 16 |
coefficient | Std. Error | t | P-value | ||
(Constant) | 660.70 | 139.247 | 4.745 | .000 | |
Depth | -52.218 | 13.589 | -3.843 | .002 |
Three-way ANOVA Test
Null hypothesis: there is no relationship between the independent variable (elevation feet, season and peaks) interactions
Alternative hypothesis: there is relationship between the independent variable (elevation feet, season and peaks) interactions
The most appropriate test for this question is the three way Anova test. This is because there are three different variables of interest and we want to know the interaction effect between the three variable. To determine the whether we have a statistically significant three-way interaction, we need to consult the “elevation”, “season”, “peaks” row in the Test of Between-subjects effect table as shown below.
Source | Type III Sum of Squares | df | Mean Square | F | Sig. | Partial Eta Squared |
Corrected Model | 117142.996^{a} | 35 | 3346.943 | 1367.016 | .000 | .997 |
Intercept | 195261.735 | 1 | 195261.735 | 79752.179 | .000 | .998 |
Elevationfeet | 31883.951 | 2 | 15941.976 | 6511.298 | .000 | .989 |
Season | 3917.200 | 1 | 3917.200 | 1599.931 | .000 | .917 |
Peaks | 34467.387 | 5 | 6893.477 | 2815.553 | .000 | .990 |
Elevationfeet * Season | 644.600 | 2 | 322.300 | 131.639 | .000 | .646 |
Elevationfeet * Peaks | 5645.555 | 10 | 564.556 | 230.586 | .000 | .941 |
Season * Peaks | 34447.424 | 5 | 6889.485 | 2813.923 | .000 | .990 |
Elevationfeet * Season * Peaks | 6136.878 | 10 | 613.688 | 250.653 | .000 | .946 |
Error | 352.563 | 144 | 2.448 | |||
Total | 312757.294 | 180 | ||||
Corrected Total | 117495.560 | 179 |
Multiple Comparisons | ||||||
(I) Elevation(feet) | (J) Elevation(feet) | Mean Difference (I-J) | Std. Error | Sig. | 95% Confidence Interval | |
Lower Bound | Upper Bound | |||||
4000 | 6000 | -16.1303^{*} | .28568 | .000 | -16.6950 | -15.5657 |
8000 | -32.6000^{*} | .28568 | .000 | -33.1647 | -32.0353 | |
6000 | 4000 | 16.1303^{*} | .28568 | .000 | 15.5657 | 16.6950 |
8000 | -16.4697^{*} | .28568 | .000 | -17.0343 | -15.9050 | |
8000 | 4000 | 32.6000^{*} | .28568 | .000 | 32.0353 | 33.1647 |
6000 | 16.4697^{*} | .28568 | .000 | 15.9050 | 17.0343 | |
Dependent Variable: Weight | ||||||
LSD |
The post hoc test is used to confirm where the difference occurred between groups (elevation, season, peak). The table above shows that there is a statistically significant difference between all variable (elevation, season and peak) because the p-value of all the interaction are less at 0.05 at 5% level of significance.
Sum of Squares | df | Mean Square | F | P-value | ||
Regression | 0.874 | 1 | 0.874 | 77.149 | .000 | |
Residual | 0.589 | 52 | 0.011 | |||
Total | 1.464 | 53 |
coefficient | Std. Error | t | P-value | ||
(Constant) | 4.465 | 0.094 | 47.709 | .000 | |
Age | 0.015 | 0.002 | 8.783 | .000 |
The best regression equation is given as . The test shows that the variable (Age) is significant at 5% level of significance. The variable Age is significant because the p-value is less than 0.05, therefore we conclude that is it significant. The R square which is used to show the variability in AIC concentration is 0.597, which implies that the variable was able to explain about 60% of the AIC concentration.
Residual Plot
Residuals Statistics | |||||
Minimum | Maximum | Mean | Std. Deviation | N | |
Predicted Value | 5.0910 | 5.4635 | 5.2772 | .12844 | 54 |
Residual | -.20898 | .20454 | .00000 | .10545 | 54 |
Std. Predicted Value | -1.450 | 1.450 | .000 | 1.000 | 54 |
Std. Residual | -1.963 | 1.921 | .000 | .991 | 54 |
Dependent Variable: A1C |
The table above shows the residual analysis of the dependent variable with mean and standard deviation of the residual and predicted value.
Step-wise Linear Regression
Null hypothesis: there is no association between the dependent variable (Exact protein) and the independent variable (Protein 1, Protein 2, Protein 3, and Protein 4)
Alternative hypothesis: there is association between the dependent variable (Exact protein) and the independent variable (Protein 1, Protein 2, Protein 3, and Protein 4)
The appropriate test for this question is the stepwise linear regression (forward). This was chosen because we are interested in the final model among all model from the regression analysis.
Model | R | R Square | Adjusted R Square | Std. Error of the Estimate | |||
R Square Change | Sig. F Change | ||||||
1 | .897^{a} | .805 | .796 | 8.768 | .805 | .000 | |
2 | .966^{b} | .933 | .927 | 5.251 | .128 | .000 | |
3 | .981^{c} | .962 | .956 | 4.072 | .029 | .001 |
The table above shows the model summary, model 1 has an Adjusted R square of 0.796, model 2 has 0.927, while model 3 has an Adjusted R square of 0.962. It also shows that there is significant change in the Adjusted R square as the three model are statistically significant.
ANOVA | ||||||
Model | Sum of Squares | df | Mean Square | F | Sig. | |
1 | Regression | 7285.977 | 1 | 7285.977 | 94.782 | .000^{b} |
Residual | 1768.023 | 23 | 76.871 | |||
Total | 9054.000 | 24 | ||||
2 | Regression | 8447.343 | 2 | 4223.671 | 153.168 | .000^{c} |
Residual | 606.657 | 22 | 27.575 | |||
Total | 9054.000 | 24 | ||||
3 | Regression | 8705.803 | 3 | 2901.934 | 175.018 | .000^{d} |
Residual | 348.197 | 21 | 16.581 | |||
Total | 9054.000 | 24 | ||||
a. Dependent Variable: Exact[Protein] | ||||||
b. Predictors: (Constant), Method 3[Protein] | ||||||
c. Predictors: (Constant), Method 3[Protein], Method 1[Protein] | ||||||
d. Predictors: (Constant), Method 3[Protein], Method 1[Protein], Method 4[Protein] |
Coefficients | ||||||
Model | Unstandardized Coefficients | Standardized Coefficients | t | Sig. | ||
B | Std. Error | Beta | ||||
1 | (Constant) | -106.133 | 20.447 | -5.191 | .000 | |
Method 3[Protein] | 1.968 | .202 | .897 | 9.736 | .000 | |
2 | (Constant) | -127.596 | 12.685 | -10.059 | .000 | |
Method 3[Protein] | 1.823 | .123 | .831 | 14.814 | .000 | |
Method 1[Protein] | .348 | .054 | .364 | 6.490 | .000 | |
3 | (Constant) | -124.200 | 9.874 | -12.578 | .000 | |
Method 3[Protein] | 1.357 | .152 | .619 | 8.937 | .000 | |
Method 1[Protein] | .296 | .044 | .310 | 6.784 | .000 | |
Method 4[Protein] | .517 | .131 | .284 | 3.948 | .001 | |
a. Dependent Variable: Exact[Protein] |
The regression equation of the final model is given as .
Residual plot for the predict value