Here I used to data file from the researcher Udara Rathnayake who undergraduate at University of Ruhuna Sri Lanka. I used R-Studio programming language for analyze this data.
- for the data set - Light_Spectrum_excel_data_set
- for the full R codes - R-codes
The researcher has used light spectrum (as treatment with 3 different levels: Red, Blue, Control) for measure germination potential how they are significantly differing.
* He has measured different responses factors. Such as,
- Germinated_seeds(%),
- Total_seedling_Length,
- Total_Length,
- Leaf_Length,
- Leaf_Width.
(df <- read.table("Light_Spectrum.csv", header = TRUE, sep = ","))## Light_Spectrum Germinated_seeds... Total_seedling_Length Total_Length
## 1 Red 98 6.9925 12.10
## 2 Red 96 6.9550 11.89
## 3 Red 100 6.8100 11.99
## 4 Blue 72 4.2950 15.55
## 5 Blue 78 4.3000 15.33
## 6 Blue 86 4.6300 15.48
## 7 Control 84 7.3600 7.76
## 8 Control 94 7.4950 7.94
## 9 Control 90 7.6150 7.40
## Leaf_Length Leaf_Width
## 1 2.15 0.98
## 2 2.26 1.08
## 3 2.14 1.14
## 4 2.98 1.19
## 5 2.82 1.24
## 6 3.01 1.31
## 7 1.84 0.78
## 8 2.17 1.02
## 9 1.71 0.97ANOVA Models
library(dplyr)
Lights <- as.factor(df$Light_Spectrum)
x <- list(
df$Germinated_seeds...,
df$Total_seedling_Length,
df$Total_Length,
df$Leaf_Length,
df$Leaf_Width
)
models <- function(x) {
ANOV <- vector("list", length(x))
for (i in seq_along(x)) {
ANOV[[i]] <- anova(lm(x[[i]] ~ Lights))
}
ANOV
}
results <- models(x)
names(results) <- c( "Germinated_seeds...", "Total_seedling_Length",
"Total_Length", "Leaf_Length", "Leaf_Width")
results## $Germinated_seeds...
## Analysis of Variance Table
##
## Response: x[[i]]
## Df Sum Sq Mean Sq F value Pr(>F)
## Lights 2 562.67 281.333 10.729 0.01043 *
## Residuals 6 157.33 26.222
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## $Total_seedling_Length
## Analysis of Variance Table
##
## Response: x[[i]]
## Df Sum Sq Mean Sq F value Pr(>F)
## Lights 2 16.1268 8.0634 387.52 4.533e-07 ***
## Residuals 6 0.1248 0.0208
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## $Total_Length
## Analysis of Variance Table
##
## Response: x[[i]]
## Df Sum Sq Mean Sq F value Pr(>F)
## Lights 2 90.518 45.259 1367.8 1.048e-08 ***
## Residuals 6 0.199 0.033
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## $Leaf_Length
## Analysis of Variance Table
##
## Response: x[[i]]
## Df Sum Sq Mean Sq F value Pr(>F)
## Lights 2 1.7050 0.85248 35.969 0.0004562 ***
## Residuals 6 0.1422 0.02370
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## $Leaf_Width
## Analysis of Variance Table
##
## Response: x[[i]]
## Df Sum Sq Mean Sq F value Pr(>F)
## Lights 2 0.15749 0.078744 9.0165 0.01556 *
## Residuals 6 0.05240 0.008733
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1So according to these results we can conclude that outputs by hypothetically as,
Null hypothesis (H0) :- There is same effect of 3 type of light specrtrum on four different factors.
Alternative hypothesis (H1) :-The effects on four different factors are not all the same. (At least on pair of light spectrum are different).
p_val <- vector("double", length(results))
concl <- vector("double", length(results))
for (i in seq_along(results)) {
p_val[[i]] <- results[[i]][[5]][1]
concl[[i]] <- ifelse(p_val[[i]] < 0.05, "Significant", "Not_Significant")
}
P_Results <- tribble(
~Factor,
"Germinated_seeds...", "Total_seedling_Length",
"Total_Length", "Leaf_Length", "Leaf_Width"
) %>%
mutate(P_Values = p_val, Conclusions = concl)
(P_Results)## # A tibble: 5 x 3
## Factor P_Values Conclusions
## <chr> <dbl> <chr>
## 1 Germinated_seeds... 0.0104 Significant
## 2 Total_seedling_Length 0.000000453 Significant
## 3 Total_Length 0.0000000105 Significant
## 4 Leaf_Length 0.000456 Significant
## 5 Leaf_Width 0.0156 SignificantIn above graph we can see 3 type of lights spectrum have significantly different on all different factors. (All p_values are less than the alpha (0.05) value)
Mean Comparisons
comparisions <- function(x) {
comp <- vector("list", length(x))
for (i in seq_along(x)) {
comp[[i]] <- TukeyHSD(aov(lm(x[[i]] ~ Lights)))
}
comp
}
mean_com <- comparisions(x)
names(mean_com) <- c( "Germinated_seeds...", "Total_seedling_Length",
"Total_Length", "Leaf_Length", "Leaf_Width")
mean_com## $Germinated_seeds...
## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = lm(x[[i]] ~ Lights))
##
## $Lights
## diff lwr upr p adj
## Control-Blue 10.666667 -2.162053 23.49539 0.0956406
## Red-Blue 19.333333 6.504614 32.16205 0.0085771
## Red-Control 8.666667 -4.162053 21.49539 0.1759680
##
##
## $Total_seedling_Length
## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = lm(x[[i]] ~ Lights))
##
## $Lights
## diff lwr upr p adj
## Control-Blue 3.0816667 2.7202901 3.4430432 0.0000007
## Red-Blue 2.5108333 2.1494568 2.8722099 0.0000017
## Red-Control -0.5708333 -0.9322099 -0.2094568 0.0068416
##
##
## $Total_Length
## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = lm(x[[i]] ~ Lights))
##
## $Lights
## diff lwr upr p adj
## Control-Blue -7.753333 -8.209044 -7.297622 0.0e+00
## Red-Blue -3.460000 -3.915711 -3.004289 1.1e-06
## Red-Control 4.293333 3.837622 4.749044 5.0e-07
##
##
## $Leaf_Length
## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = lm(x[[i]] ~ Lights))
##
## $Lights
## diff lwr upr p adj
## Control-Blue -1.0300000 -1.4156761 -0.6443239 0.0004367
## Red-Blue -0.7533333 -1.1390094 -0.3676572 0.0023508
## Red-Control 0.2766667 -0.1090094 0.6623428 0.1494900
##
##
## $Leaf_Width
## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = lm(x[[i]] ~ Lights))
##
## $Lights
## diff lwr upr p adj
## Control-Blue -0.3233333 -0.55745337 -0.08921329 0.0128978
## Red-Blue -0.1800000 -0.41412004 0.05412004 0.1221952
## Red-Control 0.1433333 -0.09078671 0.37745337 0.2247583In here we can conclude what group exactly differ from the control group. That is in hypothetically (Reject H0 when p_value is less than the alpha(0.05) value)
H0 :- mean(i) = mean(control)
H1 :- mean(i) != mean(control ) for some i, (mean(i): mean of each treatment)
According to these results we can see basically,
+Two types of factor levels are significantly different from the control group ("Total_seedling_Length", "Total_Length") and
+Two types of factor levels are not significantly different from the control group ("Germinated_seeds(%)", "Leaf_Width").
+In ("Leaf_Length") factor Blue light spectrum is signifficantly different from the control group and Red light spectrum is not.
| Factor | Light spectrum | From Control Group |
|---|---|---|
| Germinated_seeds(%) | Red | Sgnificantly Not Different |
| - | Blue | Sgnificantly Not Different |
| Total_seedling_Length | Red | Sgnificantly Different |
| - | Blue | Sgnificantly Different |
| Total_Length | Red | Sgnificantly Different |
| - | Blue | Sgnificantly Different |
| Leaf_Lengt | Red | Sgnificantly Not Different |
| - | Blue | Sgnificantly Different |
| Leaf_Width | Red | Sgnificantly Not Different |
| - | Blue | Sgnificantly Not Different |
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