HW 4 : Intro To Factorial Design :

Q1)

\[ \underline{\text{Pop. MDL}} \\ y_{ijk} = \mu + \tau_i + \beta_j + (\tau\beta)_{ij} + \epsilon_{ijk}\\ i \in[1,a] \text{ Number of Lvls} \\ j \in[1,b] \text{ Number of Lvls} \\ k \in[1,n] \text{ Number of Repitions} \]

A) Analyze Data :

## Analysis of Variance Table
## 
## Response: Finish
##                Df  Sum Sq Mean Sq F value    Pr(>F)    
## FeedRate        2 3160.50 1580.25 55.0184 1.086e-09 ***
## Depth           3 2125.11  708.37 24.6628 1.652e-07 ***
## FeedRate:Depth  6  557.06   92.84  3.2324   0.01797 *  
## Residuals      24  689.33   28.72                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Interpretation of Anova Table :
The ANOVA table indicates the Main Effects Appear to be highly significant. However, the interaction itself is also significant.

B) Diagnostics :

We are curious about the underlying assumptions. The data appears to have an average residual of 0, with approx. random scatter. The residuals themself appear to be approx. normal with some slight deviations at the tails.

C) Point Est. :

## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
## # A tibble: 3 × 2
##   FeedRate Finish_Per_FR
##   <fct>            <dbl>
## 1 0.2               81.6
## 2 0.25              97.6
## 3 0.3              104.

D) P-value :

Suppose we consider the following :

\[ \text{Sample MDL} \\ \hat{y} = \bar{y}+ F_r+D_{pth}+( F_r*D_{pth}) \]

Seen above is the sample, model where we are given the overall average ( \(\bar{y}\) ) and are curious if the main effects : \(F_r\) or \(D_{pth}\) or the interaction effect are statistically significant \(( F_r*D_{pth})\).

\[ \underline{\text{Pop. MDL}} \\ y_{ijk} = \mu + \tau_i + \beta_j + (\tau\beta)_{ij} + \epsilon_{ijk}\\ i \in[1,a] \text{ Number of Lvls} \\ j \in[1,b] \text{ Number of Lvls} \\ k \in[1,n] \text{ Number of Repitions} \]

We test this with the following Hypothesis :

\[ \text{Main Eff.} \\ H_o : \tau_1 = \tau_2 = \tau_3 = 0 \\ H_a : \tau_i \ne 0 \\ H_o : \beta_1 = \beta_2 = \beta_3 = \beta_4 = 0 \\ H_a : \beta_i \ne 0 \\ \text{Interaction Eff.} \\ H_o : (\tau\beta)_{ij} =0 \\ H_a : (\tau\beta)_{ij} \ne 0 \\ \]

##                 Pr(>F)    
## FeedRate       < 2e-16 ***
## Depth            2e-07 ***
## FeedRate:Depth 0.01797 *  
## Residuals                 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Q2)

mdl\(coefficients[mdl\)coefficients > 0

Research Questions :

Main-Effect : Does glass type or temperature affect lightoutput?

Interaction-Effect :

## Analysis of Variance Table
## 
## Response: Resp
##                        Df  Sum Sq Mean Sq F value    Pr(>F)    
## Glass_Type              2  149369   74684  205.99 3.948e-13 ***
## Temperature             2 1965241  982621 2710.26 < 2.2e-16 ***
## Glass_Type:Temperature  4  288254   72063  198.77 1.252e-14 ***
## Residuals              18    6526     363                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Call:
## lm(formula = Resp ~ Glass_Type * Temperature, data = df)
## 
## Coefficients:
##                 (Intercept)                 Glass_TypeG2  
##                    572.6667                     -19.6667  
##                Glass_TypeG3               Temperature125  
##                      0.6667                     514.6667  
##              Temperature150  Glass_TypeG2:Temperature125  
##                    810.0000                     -32.6667  
## Glass_TypeG3:Temperature125  Glass_TypeG2:Temperature150  
##                    -33.3333                     -50.0000  
## Glass_TypeG3:Temperature150  
##                   -496.6667

According to the results, both the interaction and main effects are significant.