Laboratory Exercise 5: The \(2^k\) Factorial Design/ Design Blocking and Confounding in the \(2^k\) Factorial

Problem and Data set:

The researcher is conducting an experiment to understand the joint effects of temperature and humidity on the growth of a specific plant species. The temperature has two levels: Low (20°C) and High (30°C), while humidity also has two levels: Low (40%) and High (80%). Each combination of temperature and humidity will be tested on 16 plant samples. For each combination of temperature and humidity, the researcher measures the height of 12 randomly selected plants after two weeks from the 3 different lots. The growth in centimeters were recorded as follows.

Temperature	Humidity	Lot	Plant Growth
Low	Low	1	12.5
Low	High	1	15.2
High	Low	1	14.8
High	High	1	18.3
Low	Low	2	12.8
Low	High	2	16.3
High	Low	2	13.4
High	High	2	17.9
Low	Low	3	13.0
Low	High	3	14.5
High	Low	3	14.0
High	High	3	16.5

2. Write your experimental question. Use.

Is there a significant difference in the growth of the specific plant species under different combinations of temperature and humidity, using \(\alpha = 0.05\)?

3. Construct your null and alternative hypotheses.

The hypothesis being tested are the following:

For temperature:

\(H_o:\) There is no significant main effect of temperature on the growth of the specific plant species.
\(H_a:\) There is a significant main effect of temperature on the growth of the specific plant species.

For humidity:

\(H_o:\) There is no significant main effect of humidity on the growth of the specific plant species.
\(H_a:\) There is a significant main effect of humidity on the growth of the specific plant species.

For temperature and humidity:

\(H_o:\) There is no significant interaction effect between temperature and humidity on the growth of the plant species.
\(H_a:\) There is a significant interaction effect between temperature and humidity on the growth of the plant species.

4. Fit a full factorial model with interaction to the data.

Call:
lm(formula = `Plant Growth` ~ Temperature * Humidity, data = D)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.06667 -0.36667 -0.01667  0.43333  0.96667 

Coefficients:
                           Estimate Std. Error t value Pr(>|t|)    
(Intercept)                 17.5667     0.4353  40.360 1.56e-10 ***
TemperatureLow              -2.2333     0.6155  -3.628 0.006702 ** 
HumidityLow                 -3.5000     0.6155  -5.686 0.000462 ***
TemperatureLow:HumidityLow   0.9333     0.8705   1.072 0.314917    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.7539 on 8 degrees of freedom
Multiple R-squared:  0.8922,    Adjusted R-squared:  0.8517 
F-statistic: 22.06 on 3 and 8 DF,  p-value: 0.000318

5. Is there a significant interaction effect between temperature and humidity?

Analysis of Variance Table

Response: Plant Growth
                     Df  Sum Sq Mean Sq F value    Pr(>F)    
Temperature           1  9.3633  9.3633 16.4751 0.0036387 ** 
Humidity              1 27.6033 27.6033 48.5689 0.0001162 ***
Temperature:Humidity  1  0.6533  0.6533  1.1496 0.3149166    
Residuals             8  4.5467  0.5683                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

A two-way ANOVA was conducted to examine the interaction effect between Temperature and Tumidity on plant growth. The results revealed no significant interaction effect between Temperature and Humidity, since the critical value \(F_{0.05, 1,8}=5.32 > 1.1496\) the associated F-value of Temperature:Humidity, and the corresponding p-value for the interaction term Temperature:Humidity is 0.3149166, this means that we failed to reject the null hypothesis. The interaction effect is not statistically significant in this model.

6. Is there evidence of confounding between main effects and lot effects?

Call:
lm(formula = `Plant Growth` ~ Temperature + Humidity + Lot, data = D)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.9000 -0.5417  0.1000  0.5792  0.8167 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)     18.0333     0.6290  28.670 2.37e-09 ***
TemperatureLow  -1.7667     0.4193  -4.213  0.00294 ** 
HumidityLow     -3.0333     0.4193  -7.234 8.94e-05 ***
Lot             -0.3500     0.2568  -1.363  0.21000    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.7263 on 8 degrees of freedom
Multiple R-squared:  0.8999,    Adjusted R-squared:  0.8624 
F-statistic: 23.98 on 3 and 8 DF,  p-value: 0.0002368

In this case, the p-value associated for TemperatureLow and HumidityLow is 0.00294 and 8.94e-05 respectively, meanwhile for Lot is 0.21000, which is greater than \(\alpha=0.05\) this means that not statistically significant. Since TemperatureLow and HumidityLow has a significant effect, while the p-values for Lot are not significantly low. Therefore, there is no sufficient evidence of confounding between main effects and lot effects.

7. How would you modify the design to remove confounding?

If confounding is evident in the factorial design, in order to remove or address confounding between main effects and lot effects, we can modify the design by the use of several techniques like blocking.

• Identify potential blocking factors that may contribute to variability in the response variable but are not of primary interest.
• Introduce these blocking factors into the experimental design to account for their effects.
• Fit the model with both the primary factors and the blocking factors.

However in this case, there is no sufficient evidence of confounding between main effects and lot effects thus, there is no need to modify the design.

8. Compare the precision of estimating main effects under the current vs modified design.

Since there’s no modified design, we will only state the precision of estimating main effects under the current design. Note that the standard error of the coefficients in the linear regression model is a measure of the precision of the estimates. Now, we can focus on the standard errors associated with the main effect estimates in the model. In this case, we have the standard errors for the coefficients of Temperature, Humidity, and Lot which are 0.4193, 0.4193, and 0.2568, respectively. Observe that the standard error for the main effects is small which implies a higher precision in estimating the main effect. Moreover, we notice that the model has also a small residual standard error. A smaller residual standard error indicates that the model’s predictions are closer to the actual observed values, suggesting a better fit.

9. Provide interpretations of your findings.

• The first output is the summary of a linear regression model that predicts the Plant Growth based on Temperature and Humidity. The model has an adjusted R-squared of 0.8517, which indicates that 85.17% of the variation in Plant Growth can be explained by the model.

• The second output is the analysis of variance (ANOVA) table for the same model. It shows the sum of squares, mean squares, F-value, and p-value for each predictor variable and the interaction term. The p-values for Temperature and Humidity are 0.0036387 and 0.0001162, respectively, which indicates that they are statistically significant predictors of Plant Growth.

• The third output is the summary of another linear regression model that predicts the Plant Growth based on Temperature, Humidity, and Lot. The model has an adjusted R-squared of 0.8624, which indicates that 86.24% of the variation in Plant Growth can be explained by the model.

Based from all the output we obtained, we can draw a conclusion:

A linear model and two-way ANOVA was conducted to examine the interaction effect between temperature and humidity on the growth of the specific plant species. Using \(\alpha=0.05\), the results revealed that the main effects of Temperature and Humidity are significant, with p-values of 0.0036 and 0.0001 which is less than to significance level and F-value of 16.4751 and 48.5689 which is greater than the tabulated \(F=5.32\), respectively. Thus, we reject the null hypothesis and this suggests that the main effects of Temperature and Humidity is statistically significant. This indicates that both Temperature and Humidity have a significant effect on the growth of the specific plant species. Meanwhile, the interaction effect of Temperature:Humidity is not significant, with a p-value of 0.3149 which is greater than to significance level and F-value of 1.1496 which is less than the tabulated \(F=5.32\). Therefore, we failed to reject the null hypothesis and this suggests that the interaction effect between temperature and humidity is not statistically significant. This suggests that the effect of Temperature on the growth of the specific plant species does not depend on the level of Humidity, and vice versa.

10. Give details on the syntax used to produce your answer.

First, export the data from an excel using the read_excel() function from the readxl package.

setwd("E:/Experimental Design")

D<- read_excel("E:/EXPERIMENTAL DESIGN//2^K DATA.xlsx")

To customize the table of the data, we use kable() function from the knitr package and the kableExtra package.

kable(D, format = "html") %>%
  kable_styling(full_width = FALSE) %>%
  row_spec(0, bold = TRUE, color = "black", background = "skyblue") %>%
  row_spec(1:4, background = "white")

To fit the full factorial model with interaction we use the lm() function and analyzes the results using the summary() function.

int_model <- lm(`Plant Growth` ~ Temperature * Humidity, data = D)
summary(int_model)

In order to determine if there is a significant interaction effect between temperature and humidity, we use anova() function in the model.

anova(int_model)

Furthermore, to determine if there is evidence of confounding between main effects and lot effects, we use the linear model lm() function then we add lot for the predictor.

model <- lm(`Plant Growth` ~ Temperature + Humidity + Lot, data = D)
summary(model)