Analysis of designed data and models

example
to make a cake we mix wheat, sugar and egg with different proportions.
x1 = wheat 100 g
x2= sugar 100g
egg= 2
to design an experiment following this model
 

Wheat	Sugar	egg	taste
80	80	1
120	80	1
80	120	1
120	120	1
80	80	3
120	80	3
80	120	3
120	120	3
100	100	2
100	100	2
100	100	2

MLR- multiple linear Regression
Multiple linear regression attempts to model the relationship between two or more explanatory variables(x1, x2, x3, ...)and a response variable (y) by fitting a linear equation to observed data. Every value of the independent variable x is associated with a value of the dependent variable y.

Control of raw data--giving a overview of how to control and understand the data..
at least 3 experiments are performed at ''center points'' so we will have 3 replicate plots, these replicate plots shown in the replicate plot in the software.
- small spread between replicated center points within the response interval indicates the linearity between factors and responses- x and y.
- small spread between replicated center points higher or down the range of response interval indicates the non-linearity between x and y.
- similar spread between replicates (large spread) variation in response can not distinguished from the noise. This means the replicate plots vary so much.

Histograms of response to investigate the distribution of responses, is there need for Response transfromations. Making a histogram of each response informs us if needed to transform metric of each response, the example of histogram below shows that no need to transform metric of each response.






Fitting of model to the acquired data.

R2: how much of the variation in response(y) is described by the model, the taste is varied, one model can describe the variation in taste will have R2 value high
Q2: how much of the variation in response(y) is predicted by the model, Q2>0.5, the model is considered to be good, Q2>0.9 extremely good




Anova table (Analysis of evaluation)






0.20, 009, 0.42, 0.04, 0.05, 0.06 called regression coefficients. No of regression coefficient = 6.
DF = degree of freedom
Explanation of anova table






SSTotal == 
       (3.52^2 + 3.66^2+4.74^2+5.2^2+5.38^2+5.9^2+4.36^2+4.86^2+4.73^2+4.61^2+4.68^2) = 247.206
 average(Y-observation) = 4.69
Constant= = 11*4.69^2=242.426                
SS.ttcorrected = = (3.52-4.69) ^2 + (3.66-4.69)^2+(4.74-4.69)^2+(5.2-4.69)^2+(5.38-4.69)^2+(5.9-4.69)^2+(4.36-4.69)^2+(4.86-4.69)^2+(4.73-4.69)^2+(4.61-4.69)^2+(4.68-4.69)^2 =4.7785
SS.regression = =
SS.residual = =

Effect (coefficients)
Coefficient plot
Confidence interval