Anova, and tukeyHSD Test

Before running any tests, we should first look at the data. We will create interaction plots, which will help us visualize the effect of one factor, as we move through the levels of another factor.

If there is not interaction, thus an additive model, we would expect to see parallel lines. That would mean, when we change the level of one factor, there can be an effect on the response. However, the difference between the levels of the other factor should still be the same.

The obvious indication of interaction would be lines that cross while heading in different directions. Here we don’t see that, but the lines aren’t strictly parallel, and there is some overlap on the right panel. However, is this interaction effect significant?

The mean concentration of various elements were higher under combination of blue light followed by the combination of Cocopeat + Sand and red light.

Highest concentrations are recorded for the combination of; blue light and sand followed by red light and cocopeat+sand. Concentrations were low in the combination of blue light and cocopeat.

The highest leaf area in observed plants was recorded when sand was used as media while the highest Vitamin C was recorded for Cocopeat. Likewise, higher concentrations of chlorophyl B was highest in Cocopeat+Sand while Nitrate was highest in Cocopeat. Caretinoid were high in Sand.

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Additive Anova

The row for AB Interaction tests:

\[ H_0: \text{ All }(\alpha\beta)_{ij} = 0. \quad \text{vs} \quad H_1: \text{ Not all } (\alpha\beta)_{ij} \text{ are } 0. \]

Null Model: \(y_{ijk} = \mu + \alpha_i + \beta_j + \epsilon_{ijk}.\) (Additive Model.)
Alternative Model: \(y_{ijk} = \mu + \alpha_i + \beta_j + (\alpha \beta)_{ij} + \epsilon_{ijk}.\) (Interaction Model.)

We reject the null when the \(F\) statistic is large. Under the null hypothesis, the distribution of the test statistic is \(F\) with degrees of freedom \((I -1)(J -1)\) and \(IJ(K - 1)\).

The row for Factor B tests:

\[ H_0: \text{ All }\beta_{j} = 0. \quad \text{vs} \quad H_1: \text{ Not all } \beta_{j} \text{ are } 0. \]

Null Model: \(y_{ijk} = \mu + \alpha_i + \epsilon_{ijk}.\) (Only Factor A Model.)
Alternative Model: \(y_{ijk} = \mu + \alpha_i + \beta_j + \epsilon_{ijk}.\) (Additive Model.)

We reject the null when the \(F\) statistic is large. Under the null hypothesis, the distribution of the test statistic is \(F\) with degrees of freedom \(J - 1\) and \(IJ(K - 1)\).

The row for Factor A tests:

\[ H_0: \text{ All }\alpha_{i} = 0. \quad \text{vs} \quad H_1: \text{ Not all } \alpha_{i} \text{ are } 0. \]

Null Model: \(y_{ijk} = \mu + \beta_j + \epsilon_{ijk}.\) (Only Factor B Model.)
Alternative Model: \(y_{ijk} = \mu + \alpha_i + \beta_j + \epsilon_{ijk}.\) (Additive Model.)

We reject the null when the \(F\) statistic is large. Under the null hypothesis, the distribution of the test statistic is \(F\) with degrees of freedom \(I - 1\) and \(IJ(K - 1)\).

These tests should be performed according to the model hierarchy. First consider the test of interaction. If it is significant, we select the interaction model and perform no further testing. If interaction is not significant, we then consider the necessity of the individual factors of the additive model.

Light had a significant effect on Vitamin C concentrations (p=0.003488) at p<0.01.

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
Light	3	4304.47425	1434.8248	5.7196020	0.0034879
Media	2	206.68882	103.3444	0.4119590	0.6663007
Rep	2	22.61201	11.3060	0.0450688	0.9560009
Residuals	28	7024.10635	250.8609	NA	NA

Media had a significant effect on Flavonoid concentrations (p=0.02525) at p<0.05.

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
Light	3	0.0141417	0.0047139	0.0331189	0.9917261
Media	2	1.1978167	0.5989083	4.2078221	0.0252453
Rep	2	0.0056167	0.0028083	0.0197308	0.9804762
Residuals	28	3.9853000	0.1423321	NA	NA

Media had a significant effect on Phenols concentrations (p=0.02486) at p<0.05.

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
Light	3	0.0095861	0.0031954	0.5655523	0.6422648
Media	2	0.0477722	0.0238861	4.2276303	0.0248639
Rep	2	0.0002722	0.0001361	0.0240905	0.9762176
Residuals	28	0.1582000	0.0056500	NA	NA

Light had a significant effect on Nitrate concentrations (p=0.00052) at p<0.001.

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
Light	3	5523.7931194	1841.2643731	8.0152015	0.0005200
Media	2	239.8611167	119.9305583	0.5220693	0.5989540
Rep	2	0.2703167	0.1351583	0.0005884	0.9994118
Residuals	28	6432.2029222	229.7215329	NA	NA

Light had a significant effect on Caretinoid concentrations (p=0.0361) at p<0.05.

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
Light	3	1989.669111	663.223037	3.2627939	0.0361032
Media	2	52.455039	26.227519	0.1290290	0.8794681
Rep	2	2.771572	1.385786	0.0068175	0.9932073
Residuals	28	5691.516433	203.268444	NA	NA

Media had a significant effect on Chlorophyl A concentrations (p=2.34e-06) at p<0.001.

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
Light	3	3.575632e+04	1.191877e+04	0.9142763	0.4466648
Media	2	5.565310e+05	2.782655e+05	21.3454493	0.0000023
Rep	2	1.977096e-01	9.885480e-02	0.0000076	0.9999924
Residuals	28	3.650162e+05	1.303629e+04	NA	NA

Combination of Light and Media had a significant effect on Chlorophyl B concentrations (p=0.0009425 and p=1.453e-07) at p<0.001.

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
Light	3	2.859054e+05	95301.81316	7.2660564	0.0009425
Media	2	7.634870e+05	381743.48916	29.1051097	0.0000001
Rep	2	2.428141e+00	1.21407	0.0000926	0.9999074
Residuals	28	3.672488e+05	13116.02988	NA	NA

Combination of Light and Media had a significant effect on Total Chlorophyl concentrations (p=0.013679 and p=0.0004012) at p<0.05 and p<0.001 respectively.

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
Light	3	357041.14327	119013.71442	4.2401335	0.0136799
Media	2	588106.42576	294053.21288	10.4763127	0.0004012
Rep	2	82.15413	41.07706	0.0014635	0.9985377
Residuals	28	785914.87195	28068.38828	NA	NA

Media had a significant effect on Leaf area (p=0.02424) at p<0.05.

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
Light	3	290.1275	96.70917	1.0749461	0.3756471
Media	2	766.6689	383.33444	4.2608563	0.0242380
Rep	2	134.2172	67.10861	0.7459287	0.4834847
Residuals	28	2519.0628	89.96653	NA	NA

Media had little significant effect on Plant height (p=0.02424) at p<0.1.

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
Light	3	1.8875000	0.6291667	0.7686341	0.5212754
Media	2	4.6605556	2.3302778	2.8468307	0.0749096
Rep	2	0.5088889	0.2544444	0.3108472	0.7353222
Residuals	28	22.9194444	0.8185516	NA	NA

Interaction Anova