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## Linear Regression Using Tables

This example shows how to perform linear and stepwise regression analyses using tables.

### Store predictor and response variables in a table.

tbl = table(X(:,7),X(:,8),X(:,9),X(:,15),'VariableNames',...
{'curb_weight','engine_size','bore','price'});

### Fit linear regression model.

Fit a linear regression model that explains the price of a car in terms of its curb weight, engine size, and bore.

fitlm(tbl,'price~curb_weight+engine_size+bore')
ans =
Linear regression model:
price ~ 1 + curb_weight + engine_size + bore

Estimated Coefficients:
Estimate        SE         tStat       pValue
__________    _________    _______    __________

(Intercept)        64.095        3.703     17.309    2.0481e-41
curb_weight    -0.0086681    0.0011025    -7.8623      2.42e-13
engine_size     -0.015806     0.013255    -1.1925       0.23452
bore              -2.6998       1.3489    -2.0015      0.046711

Number of observations: 201, Error degrees of freedom: 197
Root Mean Squared Error: 3.95
F-statistic vs. constant model: 136, p-value = 1.14e-47

The command fitlm(tbl) also returns the same result because fitlm, by default, assumes the response variable is in the last column of the table tbl.

### Recreate table and repeat analysis.

This time, put the response variable in the first column of the table.

tbl = table(X(:,15),X(:,7),X(:,8),X(:,9),'VariableNames',...
{'price','curb_weight','engine_size','bore'});

When the response variable is in the first column of tbl, define its location. For example, fitlm, by default, assumes that bore is the response variable. You can define the response variable in the model using either:

fitlm(tbl,'ResponseVar','price');

or

fitlm(tbl,'ResponseVar',logical([1 0 0 0]));

### Perform stepwise regression.

'ResponseVar','price')
1. Removing bore^2, FStat = 0.01282, pValue = 0.90997
2. Removing engine_size^2, FStat = 0.078043, pValue = 0.78027
3. Removing curb_weight:bore, FStat = 0.70558, pValue = 0.40195
ans =
Linear regression model:
price ~ 1 + curb_weight*engine_size + engine_size*bore + curb_weight^2

Estimated Coefficients:
Estimate          SE         tStat       pValue
___________    __________    _______    __________

(Intercept)                     131.13        14.273     9.1873    6.2319e-17
curb_weight                  -0.043315     0.0085114    -5.0891    8.4682e-07
engine_size                   -0.17102       0.13844    -1.2354       0.21819
bore                           -12.244         4.999    -2.4493      0.015202
curb_weight:engine_size    -6.3411e-05    2.6577e-05     -2.386      0.017996
engine_size:bore              0.092554      0.037263     2.4838      0.013847
curb_weight^2               8.0836e-06    1.9983e-06     4.0451    7.5432e-05

Number of observations: 201, Error degrees of freedom: 194
Root Mean Squared Error: 3.59
F-statistic vs. constant model: 89.5, p-value = 3.58e-53

The initial model is a quadratic formula, and the lowest model considered is the constant. Here, stepwiselm performs a backward elimination technique to determine the terms in the model. The final model is price ~ 1 + curb_weight*engine_size + engine_size*bore + curb_weight^2, which corresponds to

$P={\beta }_{0}+{\beta }_{C}C+{\beta }_{E}E+{\beta }_{B}B+{\beta }_{CE}CE+{\beta }_{EB}EB+{\beta }_{{C}^{2}}{C}^{2}+ϵ$

where $P$ is price, $C$ is curb weight, $E$ is engine size, $B$ is bore, ${\beta }_{i}$ is the coefficient for the corresponding term in the model, and $ϵ$ is the error term. The final model includes all three main effects, the interaction effects for curb weight and engine size and engine size and bore, and the second-order term for curb weight.