FSRaddt produces t deletion tests for each explanatory variable.

n=200; p=3; randn('state', 123456); X=randn(n,p); % Uncontaminated data y=randn(n,1); [out]=FSRaddt(y,X);

We perform a variable selection on the 'famous' stack loss data using different transformation scales for the response.

load('stack_loss'); y=stack_loss{:,4}; X=stack_loss{:,1:3}; %We start with a fan plot based on first-order model and the five most common values of ? (Figure below). [out]=FSRfan(y,X,'plots',1); %The fan plot shows that the square root transformation, ?= 0.5, is supported by all the data, with the absolute value of the statistic always less than 1.5. The evidence for all other transformations depends on which observations have been deleted: the log transformation is rejected when some of the suspected outliers are introduced into the data although it is acceptable for all the data: ?= 1 is rejected as soon as any of the suspected outliers are present. %Given that the transformation for the response which is chosen depends on the number of units declared as outliers we perform a variable selection using the original scale, the square root and the log transformation. %Robust variable selection using original untransformed values of the response % Monitoring of deletion t stat in the original scale [out]=FSRaddt(y,X,'plots',1,'quant',[0.025 0.975]); %Robust variable selection using square root values % Monitoring of deletion t stat using transformed response based on the square root [out]=FSRaddt(y.^0.5,X,'plots',1,'quant',[0.025 0.975]); %Robust variable selection using log transformed values of the response % Monitoring of deletion t stat using log transformed values [out]=FSRaddt(log(y),X,'plots',1,'quant',[0.025 0.975]); %Conclusion: the forward analysis based on the deletion t statistics clearly reveals that variable X3, independently from the transformation which is chosen and the number of outliers which are declared, is NOT significant.

Total estimated time to complete LMS: 0.01 seconds Total estimated time to complete LMS: 0.01 seconds Total estimated time to complete LMS: 0.01 seconds Total estimated time to complete LMS: 0.01 seconds Total estimated time to complete LMS: 0.01 seconds Total estimated time to complete LMS: 0.01 seconds Total estimated time to complete LMS: 0.01 seconds Total estimated time to complete LMS: 0.01 seconds ------------------------------ Warning: Number of subsets without full rank equal to 14.9% Total estimated time to complete LMS: 0.01 seconds Total estimated time to complete LMS: 0.01 seconds Total estimated time to complete LMS: 0.01 seconds ------------------------------ Warning: Number of subsets without full rank equal to 14.3% Total estimated time to complete LMS: 0.01 seconds Total estimated time to complete LMS: 0.01 seconds Total estimated time to complete LMS: 0.01 seconds ------------------------------ Warning: Number of subsets without full rank equal to 14.9%

Example of use of FSRaddt with plot of deletion t with personalized line width for the envelopes and personalized confidence interval.

n=200; p=3; X=randn(n,p); y=randn(n,1); kk=9; y(1:kk)=y(1:kk)+6; X(1:kk,:)=X(1:kk,:)+3; [out]=FSRaddt(y,X,'plots',1,'quant',[0.025 0.975]);

load('wool'); y=log(wool{:,end}); X=wool{:,1:end-1}; [out]=FSRaddt(y,X,'plots',1);

Line width equal to 3 for the curves representing envelopes; line width equal to 4 for the curves associated with deletion t stat.

n=200; p=3; randn('state', 123456); X=randn(n,p); % Uncontaminated data y=randn(n,1); [out]=FSRaddt(y,X,'plots',1,'nameX',{'F1','F2','F3'},'lwdenv',3,'lwdt',4);

Total estimated time to complete LMS: 0.02 seconds Total estimated time to complete LMS: 0.01 seconds Total estimated time to complete LMS: 0.02 seconds

`y`

— Response variable.
Vector.A vector with n elements that contains the response variable. It can be either a row or a column vector.

**
Data Types: **`single| double`

`X`

— Predictor variables.
Matrix.Data matrix of explanatory variables (also called 'regressors') of dimension (n x p-1). Rows of X represent observations, and columns represent variables.

Missing values (NaN's) and infinite values (Inf's) are allowed, since observations (rows) with missing or infinite values will automatically be excluded from the computations.

**
Data Types: **`single| double`

Specify optional comma-separated pairs of `Name,Value`

arguments.
`Name`

is the argument name and `Value`

is the corresponding value. `Name`

must appear
inside single quotes (`' '`

).
You can specify several name and value pair arguments in any order as ```
Name1,Value1,...,NameN,ValueN
```

.

```
'intercept',false
```

,```
'h',round(n*0,75)
```

,```
'nsamp',1000
```

,```
'lms',1
```

,```
'init',100 starts monitoring from step m=100
```

,```
'plots',1
```

,```
'nameX',{'NameVar1','NameVar2'}
```

,```
'lwdenv',1
```

,```
'quant',[0.025 0.975]
```

,```
'lwdt',1
```

,```
'nocheck',true
```

,```
'titl','Example'
```

,```
'labx','Subset'
```

,```
'laby','statistics'
```

,```
'FontSize',11
```

,```
'SizeAxesNum',11
```

,```
'ylimy',[0 1]
```

,```
'xlimy',[0 1]
```

`intercept`

—Indicator for constant term.true (default) | false.Indicator for the constant term (intercept) in the fit, specified as the comma-separated pair consisting of 'Intercept' and either true to include or false to remove the constant term from the model.

**Example: **```
'intercept',false
```

**Data Types: **`boolean`

`h`

—The number of observations that have determined the
least trimmed squares estimator.scalar.h is an integer greater or equal than [(n+size(X,2)+1)/2] but smaller then n

**Example: **```
'h',round(n*0,75)
```

**Data Types: **`double`

`nsamp`

—Number of subsamplse which will be extracted to find the
robust estimator.scalar.If nsamp=0 all subsets will be extracted. They will be (n choose p). Remark: if the number of all possible subset is <1000 the default is to extract all subsets otherwise just 1000.

**Example: **```
'nsamp',1000
```

**Data Types: **`double`

`lms`

—Criterion to use to find the initlal subset to
initialize the search.scalar, vector | structure.If lms=1 (default) Least Median of Squares is computed, else Least Trimmed of Squares is computed. else (default) no plot is produced

**Example: **```
'lms',1
```

**Data Types: **`double`

`init`

—Search initialization.scalar.Scalar which specifies the initial subset size to start monitoring exceedances of minimum deletion residual, if init is not specified it will be set equal to:

p+1, if the sample size is smaller than 40;

min(3*p+1,floor(0.5*(n+p+1))), otherwise.

**Example: **```
'init',100 starts monitoring from step m=100
```

**Data Types: **`double`

`plots`

—Plot on the screen.scalar.If plots=1 a plot with forward deletion t-statistics is produced

**Example: **```
'plots',1
```

**Data Types: **`double`

`nameX`

—Add variable labels in plot.cell array of strings.Cell array of strings of length p containing the labels of the varibles of the regression dataset. If it is empty (default) the sequence X1, ..., Xp will be created automatically

**Example: **```
'nameX',{'NameVar1','NameVar2'}
```

**Data Types: **`cell`

`lwdenv`

—Line width for envelopes.scalar.Line width for envelopes based on student T (default is 2)

**Example: **```
'lwdenv',1
```

**Data Types: **`double`

`quant`

—Confidence quantiles for the envelopes.vector.Confidence quantiles for the envelopes of deletion t stat. Default is [0.005 0.995] (i.e. a 99% pointwise confidence interval)

**Example: **```
'quant',[0.025 0.975]
```

**Data Types: **`double`

`nocheck`

—Check input arguments.boolean.If nocheck is equal to true no check is performed on matrix y and matrix X. Notice that y and X are left unchanged. In other words the additional column of ones for the intercept is not added. As default nocheck=false.

**Example: **```
'nocheck',true
```

**Data Types: **`boolean`

`labx`

—a label for the x-axis.character.(default: 'Subset size m')

**Example: **```
'labx','Subset'
```

**Data Types: **`char`

`laby`

—a label for the y-axis.character.(default: 'Deletion t statistics')

**Example: **```
'laby','statistics'
```

**Data Types: **`char`

`FontSize`

—the font size of the labels of
the axes and of the labels inside the plot.scalar.Default value is 12

**Example: **```
'FontSize',11
```

**Data Types: **`double`

`SizeAxesNum`

—size of the numbers of the axes.scalar.Default value is 10

**Example: **```
'SizeAxesNum',11
```

**Data Types: **`double`

`ylimy`

—minimum and maximum of the y axis.vector.Default value is '' (automatic scale)

**Example: **```
'ylimy',[0 1]
```

**Data Types: **`double`

`xlimx`

—minimum and maximum of the x axis.vector.Default value is '' (automatic scale)

**Example: **```
'xlimy',[0 1]
```

**Data Types: **`double`

`out`

— description
StructureStructure which contains the following fields

Value | Description |
---|---|

`Tdel` |
(n-init+1) x (p+1) matrix containing the monitoring of deletion t stat in each step of the forward search 1st col = fwd search index (from init to n) 2nd col = deletion t stat for first explanatory variable 3rd col = deletion t stat for second explanatory variable ... (p+1)th col = deletion t stat for pth explanatory variable |

`S2del` |
(n-init+1) x (p+1) matrix containing the monitoring of deletion t stat in each step of the forward search 1st col = fwd search index (from init to n) 2nd col = deletion t stat for first explanatory variable 3rd col = deletion t stat for second explanatory variable ... (p+1)th col = deletion t stat for pth explanatory variable |

`Una` |
cell of size p. out.Una{i} (i=1, ..., p) is a (n-init) x 11 matrix which contains the unit(s) included in the subset at each step in the search which excludes the ith explanatory variable. REMARK: in every step the new subset is compared with the old subset. Un contains the unit(s) present in the new subset but not in the old one Un(1,:) for example contains the unit included in step init+1 ... Un(end,2) contains the units included in the final step of the search |

Atkinson, A.C. and Riani, M. (2002b), Forward search added variable t tests and the effect of masked outliers on model selection, "Biometrika", Vol. 89, pp. 939-946.