Study Of Differential Effects Of Rhizomes: Research Proposal And Methodology

Research Questions

Question:

Write a report about the study of differential effects of Rhizomes. 

Sometimes when we are dealing with many factors and with a large number of experimental units, sometimes, an observation may miss due to various reasons such as carelessness, error, for reasons beyond control such as damage to an experimental unit. If there is a missing observation in RBD, randomized block design, every treatment does not occur in every block. Then, treatments are no longer orthogonal to blocks. The study of the differential effects of Rhizomes is very useful for the understanding of the nature of the rhizomes and the effects associated with the rhizomes. There are so many significant interaction produces due to the differential effects of the Rhizomes. We want to study these significant effects due to the Rhizomes. Let us see this research proposal in detail explained in the next topic.

In the article “Do rhizome severing and shoot defoliation affect clonal growth of Leymus chinensis at ramet population level?” by Zhengwen Wang (2004), it is said that Leymus chinensis Tzvel is a perennial species of Gramineae usually subject to the defoliation from grazing and mowing. Researcher examine whether shoot defoliation and rhizome severing affected rhizome and ramet growth. It is observed that the defoliation affected the rhizome growth and the bud outgrowth. It is also observed that there is a little effect on shoot growth. There is a similar effect of the mild and heavy defoliation on the rhizome growth. It was observed that only heavy defoliation is significantly reduces the bud outgrowth while the mild defoliation did not reduces significantly. It is also observed the fact that shoot growth did not change after defoliation and that the bud numbers remained unchanged after mild defoliation suggest that the compensatory response enable the species to tolerate grazing to some extent. Neither rhizome severing nor the interaction of rhizome severing and defoliation had effect on any tested variables. Lack of the effect of rhizome severing falsified the first half of our hypothesis, that is, clonal integration was unimportant in our experiment. The probable reasons were suspected to be the short duration of the experiment and/or the buffer effect of carbohydrate reserves in rhizomes for shoot growth and bud production in time of defoliation.

Suppose we have data observed from an experiment using the treatment Biltox, Dithane, Brestan, Control. After sowing rhizomes of the mat grass in four plots in each of the three villages the above four treatments were applied at random to the plots in a village after 30 days of sowing. Unfortunately the yield of the third treatment in the third village was lost.

It is important to establish the research questions for the research process because it provides the proper way to the research study. For this research study, we have to check the different claims regarding the differential effects of rhyzomes. The research questions for this research proposal are summarised as below:

• Here we want to estimate the unknown value corresponding to Bresnan.
• Analyze the data to find out if there are any significant treatment effects
• Whether the blocking effect was important or not.

To answers these questions, it is important to collect the data from the differential effects of rhyzomes and use the proper research methodology for drawing conclusions. Let us see the research methodology used for this research study. 

Research Methodology

The first step in the research methodology for the business research study is nothing but to establish the research hypothesis or the research questions. By establishing the research questions we get the proper guideline for the research process. After establishing the research questions we need to collect the data for the variables included in the research study. For this research study we have to collect the data regarding the differential effects of rhyzomes. The given layout is that of an RBD, randomized block design where there are two sources of variation one along the treatments and one along the blocks. Let us first thoroughly go through the theory of randomized block design. Randomized block design is the simplest design that implements all the three main principles of design – randomization, replication and local control. Here the experimental units can be heterogeneous with respect to some nuisance factors. Here we group the units into blocks so that the units within a block are homogeneous. In a randomized block design randomization is restricted within a block. 

There are various methods of missing data analysis one such method is Yates Method.

Let one of the observations be missing denoted as X

1. We write down the sum of squares of error in terms of x and the available data. This expression is usually a quadratic in x say SSE(x).The estimate of x is obtained as that value of x* for which SSE(x*) is minimum.SSE(x*) is the corrected SSE with degrees of freedom ve-1,where ve is the degrees of freedom of SSE had there been no missing values.
2. The treatment sum of squares and the error sum of squares are written as the function of X and the available data.
3. The estimate of X is obtained as X** for which SSE(x**) + SSE(x**) is minimum. This is the corrected sum of squares due to treatments with DF vt + ve-1 where vt is the df of SST had there been no missing observations.
4. To test the differential effects of the treatments the sum of squares due to the hypothesis is SSHo=SST(x**) + SSE(x**)-SSE(x*) which has degrees of freedom vt.  

We know that the data collection is very important for the purpose of carrying out the research process. After sowing rhizomes of the mat grass in four plots in each of the three villages the above four treatments were applied at random to the plots in a village after 30 days of sowing. Unfortunately the yield of the third treatment in the third village was lost. The available yields in gm of 1 rgft. cutting per plot after 120 days are given below.

Treatment		Village
	1	2	3
Biltox	678	10	531
Dithane	703	689	611
Bresnan	736	574	*
Control	556	510	500

Here we use missing plot technique as we have one data point missing. This algorithm is suggested by Yates. Here at first we write down the SSE as a quadratic in x. The estimate of x is X* which is the value of X for which SSE is minimum SSE then from SSE(X*) we get the corrected SSE. Next we write SST + SSE in quadratic of x then we find X** which is the value of X for which SST+SSE is minimum. This is the corrected sum of squares due to the treatments. 

Here the given layout is that of an RBD (Randomized Block Design) with 3 villages as 3 blocks and 4 treatments.

Let, y_ij denote the response variable of I th village and jth treatment.

Here y₃₃ is missing.

Let,y₃₃=x

B₃=sum of available observations in 3 rd block=1642

T₃= sum of available observations in 3 rd treatment=1310

G= sum of all available observations=6298

Here r= number of blocks=3

v= number of treatments=4

Now x==x^*=594.667

SSE(X*)=11535.9 with degrees of freedom 6

Now, X**==547.33

SSHO(X**,X*)=SST(X**)+SSE(X**)-SSE(X*)=34892.3

Suppose we want to test the differential effects of the treatments i.e. whether the yield of mat grass depends on the treatments or not .

We want to test Hot:T₁=…=T₄=0 against H1t:Not Hot

The test statistic is given by

F_t=*()  ~ F_v-1,rv-r-v

Reject Hot against H1t if obs. (F_t)> F_{v-1,rv-r-v,alpha} 

A project plan using a Gantt chart

A Gantt chart is the chart of the proper time management for the different activities included in the research study. For this research study we have to perform different activities for the purpose of data collection and data analysis. The general Gantt chart for this research project is given as below:

Activity	Day
1	2	3	4	5	6
Study of research topic
Data collection – Experiment
Data collection – Readings
Checking assumptions
Handling problem with missing data entries
Statistical data analysis by using Minitab or SPSS
Research outcomes

For this research study, the research outcomes are summarised as below:

Now if we take level of significance=alpha=0.05

Then F_{v-1,rv-r-v,alpha}=6.59138

Obs.(F_t)=5.04112

So we can conclude that Ho is accepted at 5% level of significance hence the treatment effects do not differ significantly. We can also test whether there is any difference in the blocks or not, if the block effects are significantly different then we might have succeeded in achieving local control. But if block effects are not significantly different then there may be 2 cases, firstly our data was already homogeneous so there was no need of blocking otherwise we have failed to make the blocks. If we have succeeded in making the blocking effect significantly different there may be the case of the variability due to error which can be easily solved by just modifying the Model.  

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