Experimental Design

Completely Random Design

The response is the length of odontoblasts (cells responsible for tooth growth) in 60 guinea pigs. Each animal received one of three dose levels of vitamin C (0.5, 1, and 2 mg/day) by one of two delivery methods, orange juice or ascorbic acid (a form of vitamin C and coded as VC).

  • Treatment Structure: 2 x 3 Factorial Treatment, both Fixed

  • Model: \(y_{ijk} = \mu + \alpha_i + \beta_j + \alpha \beta_{ij} + e_{ijk}\)
  • Treatments: \(\alpha_i = \text{supp, } \beta_j = \text{dose}\)
  • Fixed Effects: \(\alpha_1 = \beta_1 = \alpha \beta_{1j} = \alpha \beta_{i1} = 0\)

  • Random Effects: \(e_{ijk} = N(0, \sigma^2_e)\)

      len        supp         dose       dose.factor
 Min.   : 4.20   OJ:30   Min.   :0.500   0.5:20     
 1st Qu.:13.07   VC:30   1st Qu.:0.500   1  :20     
 Median :19.25           Median :1.000   2  :20     
 Mean   :18.81           Mean   :1.167              
 3rd Qu.:25.27           3rd Qu.:2.000              
 Max.   :33.90           Max.   :2.000              
    
     0.5  1  2
  OJ  10 10 10
  VC  10 10 10
Analysis of Variance Table

Response: len
                 Df  Sum Sq Mean Sq F value    Pr(>F)    
supp              1  205.35  205.35  15.572 0.0002312 ***
dose.factor       2 2426.43 1213.22  92.000 < 2.2e-16 ***
supp:dose.factor  2  108.32   54.16   4.107 0.0218603 *  
Residuals        54  712.11   13.19                      
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
 supp dose.factor lsmean       SE df  lower.CL upper.CL
 OJ   0.5          13.23 1.148353 54 10.927691 15.53231
 VC   0.5           7.98 1.148353 54  5.677691 10.28231
 OJ   1            22.70 1.148353 54 20.397691 25.00231
 VC   1            16.77 1.148353 54 14.467691 19.07231
 OJ   2            26.06 1.148353 54 23.757691 28.36231
 VC   2            26.14 1.148353 54 23.837691 28.44231

Confidence level used: 0.95 
  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = mdl)

$supp
      diff       lwr       upr     p adj
VC-OJ -3.7 -5.579828 -1.820172 0.0002312

$dose.factor
        diff       lwr       upr   p adj
1-0.5  9.130  6.362488 11.897512 0.0e+00
2-0.5 15.495 12.727488 18.262512 0.0e+00
2-1    6.365  3.597488  9.132512 2.7e-06

$`supp:dose.factor`
               diff        lwr        upr     p adj
VC:0.5-OJ:0.5 -5.25 -10.048124 -0.4518762 0.0242521
OJ:1-OJ:0.5    9.47   4.671876 14.2681238 0.0000046
VC:1-OJ:0.5    3.54  -1.258124  8.3381238 0.2640208
OJ:2-OJ:0.5   12.83   8.031876 17.6281238 0.0000000
VC:2-OJ:0.5   12.91   8.111876 17.7081238 0.0000000
OJ:1-VC:0.5   14.72   9.921876 19.5181238 0.0000000
VC:1-VC:0.5    8.79   3.991876 13.5881238 0.0000210
OJ:2-VC:0.5   18.08  13.281876 22.8781238 0.0000000
VC:2-VC:0.5   18.16  13.361876 22.9581238 0.0000000
VC:1-OJ:1     -5.93 -10.728124 -1.1318762 0.0073930
OJ:2-OJ:1      3.36  -1.438124  8.1581238 0.3187361
VC:2-OJ:1      3.44  -1.358124  8.2381238 0.2936430
OJ:2-VC:1      9.29   4.491876 14.0881238 0.0000069
VC:2-VC:1      9.37   4.571876 14.1681238 0.0000058
VC:2-OJ:2      0.08  -4.718124  4.8781238 1.0000000

Are the necessary conditions for hypothesis testing present?

  • Normality: Residuals appear normally distributed per the residual normal reference plot and shapiro-wilks test
  • Equal Variance: Brown-Forsythe-Levene test and residual plot supports equal variance
  • Independence: No correlation in the residuals per the Durbin Watson test and plots of variables against residuals

Conditions for hypothesis testing appears to be satisfied


    Shapiro-Wilk normality test

data:  mdl$residuals
W = 0.98499, p-value = 0.6694

Levene's Test for Homogeneity of Variance (center = median)
      Df F value Pr(>F)
group  5  1.7086 0.1484
      54               

 lag Autocorrelation D-W Statistic p-value
   1     -0.02932541      2.025437   0.586
 Alternative hypothesis: rho != 0
  • Group doses so that each dose is not statistically different than any other dose in the group:
    • The interaction between dose and supp are significant so we need to assess the differences in dose per each level of supp.
    • OJ: {.5}, {1, 2}
    • VC: {.5}, {1}, {2}
  • Group supps so that each supp is not statistically different than any other supp in the group:
    • The interaction is significant so we need to assess the supps at each level of dose
    • .5: {OJ}, {VC}
    • 1: {OJ}, {VC}
    • 2: {OJ, VC}

Random Complete Block Design

An experiment was conducted to compare four different pre-planting treatments for soybeen seeds. A fifth treatment, consisting of not treating the seeds was used as a control. The experimental area consisted of four fields. There are notable differences in the fields. Each field was divided into five plots and one of the treatments was randomly assigned to a plot within each field.

  • Treatment Structure: 1 Single Treatment with 5 levels

  • Response: The number of plants that failed to emerge out of 100 seeds planted per plot.

  • Model: \(y_{ij} = \mu + \alpha_i + \beta_j + e_{ij}\)
  • Treatments: \(\alpha_i = \text{Seed, } \beta_j = \text{Field, } \alpha_5 = \text{Control}\)
  • Fixed Effects: \(\alpha_5 = \beta_1 = 0\)

  • Random Effects: \(e_{ij} = N(0, \sigma^2_e)\)

Soy Bean Data
Treatment Field.1 Field.2 Field.3 Field.4
Avasan 2 5 7 11
Spergon 4 10 9 8
Semaesan 3 6 9 10
Fermate 9 3 5 5
Control 8 11 12 13 
Analysis of Variance Table

Response: Count
          Df Sum Sq Mean Sq F value  Pr(>F)  
Field      3   49.8 16.6000  2.5971 0.10070  
Treatment  4   72.5 18.1250  2.8357 0.07227 .
Residuals 12   76.7  6.3917                  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

0.0.0.1 Comparison of means vs control

Since we have a control variable we want to know if any of the treatment means are significantly lower than the control mean.


     Simultaneous Tests for General Linear Hypotheses

Multiple Comparisons of Means: Dunnett Contrasts


Fit: lm(formula = Count ~ Field + Treatment, data = soy)

Linear Hypotheses:
                        Estimate Std. Error t value Pr(<t)  
Avasan - Control >= 0     -4.750      1.788  -2.657 0.0325 *
Fermate - Control >= 0    -5.500      1.788  -3.077 0.0158 *
Semaesan - Control >= 0   -4.000      1.788  -2.238 0.0668 .
Spergon - Control >= 0    -3.250      1.788  -1.818 0.1307  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Adjusted p values reported -- single-step method)

We have significant evidence that only Avasan and Fermate are significantly lower than the control. Are they significantly different from each other?

  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = mdl)

$Field
                diff        lwr      upr     p adj
Field.2-Field.1  1.8 -2.9471482 6.547148 0.6814523
Field.3-Field.1  3.2 -1.5471482 7.947148 0.2406905
Field.4-Field.1  4.2 -0.5471482 8.947148 0.0895218
Field.3-Field.2  1.4 -3.3471482 6.147148 0.8173180
Field.4-Field.2  2.4 -2.3471482 7.147148 0.4666374
Field.4-Field.3  1.0 -3.7471482 5.747148 0.9219188

$Treatment
                  diff        lwr       upr     p adj
Avasan-Control   -4.75 -10.448139 0.9481388 0.1206718
Fermate-Control  -5.50 -11.198139 0.1981388 0.0603205
Semaesan-Control -4.00  -9.698139 1.6981388 0.2305921
Spergon-Control  -3.25  -8.948139 2.4481388 0.4074833
Fermate-Avasan   -0.75  -6.448139 4.9481388 0.9926478
Semaesan-Avasan   0.75  -4.948139 6.4481388 0.9926478
Spergon-Avasan    1.50  -4.198139 7.1981388 0.9131542
Semaesan-Fermate  1.50  -4.198139 7.1981388 0.9131542
Spergon-Fermate   2.25  -3.448139 7.9481388 0.7194742
Spergon-Semaesan  0.75  -4.948139 6.4481388 0.9926478

There is not significant evidence between the difference in means between any of the treatments.