Quantitative methods can be described as an investigation of phenomena through the gathering of quantifiable data and performing mathematical and statistical techniques. Quantitative methods collect data using sampling methods. It focuses on the collection of data before generalizing it in groups to explain a specific phenomenon.

## Linear model

y_i = β_0+β_1 x_1+β_2 x_2+ β_3 x_3+ ε where y the dependent variable is the egg size, x_1 is the weight, x_2 is the Age, x_3 is the interaction effect between weight and Age and ε is the error term such that ε ~ N(0,σ^2). The parametersβ_0, β_1,β_2 and β_3 are the intercept, coefficient for x_1, coefficient for x_2and coefficient for x_3 respectively.
The multiple linear models stated above is a regression model for the variables under investigation. However, the ANOVA version of this model is a 2-way ANOVA with interaction given by the formula below.
y = μ+a_i+b_j+ 〖ab〗_ij+ ε where y is the response variable, μ is the overall mean, a_i is the ith weight, b_j is the jth Age and 〖ab〗_ij is interaction from ith weight and jth Age.
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### Visualization of the output of the linear model

The anova model fitted to the data is visualized in R and code used is presented below:
eggsize=c(15,16,15,18,22,25,28,24,23);weight=c(12,11,14,22,40,33,21,25,24)
Age=c("young","young","young","old","old","old","middle","middle","middle")
mydata=data.frame(eggsize,weight,Age)
regmodel=lm(eggsize~weight+Age,data=mydata);
anovamodel=lm(eggsize~weight+Age,data=mydata);
par(mfrow=c(2,2))
plot(regmodel) # note that either regmodel or anovamodel can be plotted and both return the same answer
require(lsmeans) # this is the package needed for Least Square methods
LSMeans=lsmeans(regmodel,"Age");
plot(LSMeans)  ### Model explanation

eggsize=51.9231–1.1538*weight–38.4615*Ageold-44.2088*Ageyoung+ 1.4130*weight:Ageold + 1.5824*weight:Ageyoung
Weight: Ageold mean Interaction effect between weight and Ageold, we are actually testing to know whether the age of the mother (old) has an influence on the size of the fish eggs.
The output of means is the averages of the predictions for each Age, over the 3 groups (young, Middle, and Old). These values often called adjusted means are not the same as the overall means for each Age group. There are means for the levels of Age which are adjusted for means of weight in the model.
From the results obtained, the egg size of young, middle, and oldfish is 20.33, 26.03, and 19.28 respectively. It can be deduced that eggs laid by middle-aged fish is the largest in size while eggs laid by youngand old aged fish are roughly the same in sizeSize of the egg gets larger as they grow from young to middle but the size drops as they get older
We use the model selected in exercise 1c to calculate the egg size of a young female fish whose weight is 17
eggsize= 51.9231-1.1538*weight-38.4615*Ageold-41.2088*Ageyoung+ 1.4130*weight:Ageold + 1.5824*weight:Ageyoung
Note, the Ageold = 0 since the prediction is for only young while Ageyoung =1
weight:Ageold = 17*0 => 0 and weight:Ageyoung = 17*1 => 17
eggsize=51.9231-1.1538*17-41.2088*1+1.5824*17
eggsize = 18.0005

### Sampling in quantitative analysis

a, An effect in a model is said to be random if the levels of the factor observed are sampled from the population of levels. In this study, subpopulation is a random effect.
The output reveals the variability across the two subpopulations as well as the significance of age (young, Middle, and Old)in the least-squares Means table.
b, Here, we have a mixed effect model with one random effect (subpopulation) and two fixed effects (Age and weight) given by:
y = μ+a_i+b_j+ c_k+ ε
c, Looking at the lsmeans in exercise 1 and the Least square means table in exercise 2, it can be observed that the adjusted means for each level of Age in exercise 2 are slightly different from the ones in exercise 1 due to the fact that a random effect (subpopulation) was added to the model earlier produced in exercise 1.
d,
regmodel=lm(eggsize~weight+factor(Age)+factor(subpopulation),data=marinedata)
summary(regmodel)
the output below is the result obtained from the R code above. Since subpopulation as a randon effect was added to the data, we update the data and produce a different model that will capture subpopulation.
Coefficients:
 Estimate Std. Error t value Pr(>|t|) (Intercept) -5.6261 14.0269 -0.401 0.6960 weight 1.2414 0.6393 1.942 0.0782 Ageold 23.7101 14.6647 1.617 0.1342 Ageyoung 11.4926 19.5055 0.589 0.5676 subpopulationB -4.1413 2.2496 -1.841 0.0927 weight:Ageold -1.1515 0.6592 -1.747 0.1085 weight:Ageyoung -0.2257 1.2167 -0.186 0.8562

The model is eggsize = -5.6261+1.2414*weight +23.7101*Ageold +11.4926*Ageyoung-4.1413*subpopulationB-1.1515*weight:Ageold 0.2257*weight:Ageyoung
eggsize = -5.6261+1.2414*17+11.4926*1-4.1413*1-0.2257*17
eggsize = 18.9921≈ 19
Hence, the egg size of young female fish with weight if 17 is 19

### Sampling in Linear model

In this study, we have 2 climates (warm and cold) of which each contains three 3 populations. The population is a dependent variable while both climate which is a dummy variable and number of diatoms are the explanatory variables in the study. Interaction effect between climate and number of diatoms will be considered to check if both influence the activity of copepods measured.
While the activity of copepods measured is continuous in nature, the number of diatoms is categorical and the climate is a dummy variable (warm or cold).
The model that best fit these data is given by:
y_i = μ+a_i+b_j+ 〖ab〗_ij+ ε this is a 2-way ANOVA with interaction
the regression equation is given by:
Y = β_0+β_1 x_1+β_2 x_2+ β_3 x_3+ ε this is a multiple linear regression model
Where y is the activity of the copepods, β_0 is the intercept, β_1 and β_2 are regression coefficients and e is the error term normally distributed mean and variance.
where y the dependent variable is the activity of the copepods, x_1 is number of diatoms, x_2 is the climate, x_3 is the interaction effect between number of diatoms andclimate and ε is the error term such that ε ~ N(0,σ^2). The parametersβ_0, β_1,β_2 and β_3 are the intercept, coefficient for x_1, coefficient for x_2and coefficient for x_3 respectively.
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