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`CovRatio`

)Delete-1 change in covariance (`CovRatio`

) identifies the
observations that are influential in the regression fit. An influential
observation is one where its exclusion from the model might significantly alter
the regression function. Values of `CovRatio`

larger than 1 +
3**p*/*n* or smaller than 1 –
3**p*/*n* indicate influential points,
where *p* is the number of regression coefficients, and
*n* is the number of observations.

The `CovRatio`

statistic is the ratio of the determinant of
the coefficient covariance matrix with observation *i* deleted
to the determinant of the covariance matrix for the full model:

$$\text{CovRatio}=\frac{\mathrm{det}\left\{MSE\left(i\right){\left[{X}^{\prime}\left(i\right)X\left(i\right)\right]}^{-1}\right\}}{\mathrm{det}\left[MSE{\left({X}^{\prime}X\right)}^{-1}\right]}.$$

`CovRatio`

is an *n*-by-1
vector in the `Diagnostics`

table of the fitted
`LinearModel`

object. Each element is the ratio of the
generalized variance of the estimated coefficients when the corresponding
element is deleted to the generalized variance of the coefficients using all the
data.

After obtaining a fitted model, say, `mdl`

, using
`fitlm`

or `stepwiselm`

, you can:

Display the

`CovRatio`

by indexing into the property using dot notationmdl.Diagnostics.CovRatio

Plot the delete-1 change in covariance using

For details, see theplotDiagnostics(mdl,'CovRatio')

`plotDiagnostics`

method of the`LinearModel`

class.

`CovRatio`

This example shows how to use the `CovRatio`

statistics to determine the influential points in data. Load the sample data and define the response and predictor variables.

```
load hospital
y = hospital.BloodPressure(:,1);
X = double(hospital(:,2:5));
```

Fit a linear regression model.

mdl = fitlm(X,y);

Plot the `CovRatio`

statistics.

`plotDiagnostics(mdl,'CovRatio')`

For this example, the threshold limits are 1 + 3*5/100 = 1.15 and 1 - 3*5/100 = 0.85. There are a few points beyond the limits, which might be influential points.

Find the observations that are beyond the limits.

find((mdl.Diagnostics.CovRatio)>1.15|(mdl.Diagnostics.CovRatio)<0.85)

`ans = `*5×1*
2
14
84
93
96

`Dfbetas`

)The sign of a delete-1 scaled difference in coefficient estimate
(`Dfbetas`

) for coefficient *j* and
observation *i* indicates whether that observation causes an
increase or decrease in the estimate of the regression coefficient. The absolute
value of a `Dfbetas`

indicates the magnitude of the difference
relative to the estimated standard deviation of the regression coefficient. A
`Dfbetas`

value larger than 3/sqrt(*n*) in
absolute value indicates that the observation has a large influence on the
corresponding coefficient.

`Dfbetas`

for coefficient *j* and
observation *i* is the ratio of the difference in the estimate
of coefficient *j* using all observations and the one obtained
by removing observation *i*, and the standard error of the
coefficient estimate obtained by removing observation *i*. The
`Dfbetas`

for coefficient *j* and
observation *i* is

$$Dfbeta{s}_{ij}=\frac{{b}_{j}-{b}_{j\left(i\right)}}{\sqrt{MS{E}_{\left(i\right)}}\left(1-{h}_{ii}\right)},$$

where
*b*_{j} is the
estimate for coefficient *j*,
*b*_{j(i)}
is the estimate for coefficient *j* by removing observation
*i*,
*MSE*_{(i)} is the
mean squared error of the regression fit by removing observation
*i*, and
*h*_{ii} is the
leverage value for observation *i*. `Dfbetas`

is an *n*-by-*p* matrix in the
`Diagnostics`

table of the fitted
`LinearModel`

object. Each cell of
`Dfbetas`

corresponds to the `Dfbetas`

value for the corresponding coefficient obtained by removing the corresponding
observation.

After obtaining a fitted model, say, `mdl`

, using
`fitlm`

or `stepwiselm`

, you can obtain
the `Dfbetas`

values as an
*n*-by-*p* matrix by indexing into the
property using dot
notation,

mdl.Diagnostics.Dfbetas

`Dfbetas`

This example shows how to determine the observations that have large influence on coefficients using `Dfbetas`

. Load the sample data and define the response and independent variables.

```
load hospital
y = hospital.BloodPressure(:,1);
X = double(hospital(:,2:5));
```

Fit a linear regression model.

mdl = fitlm(X,y);

Find the `Dfbetas`

values that are high in absolute value.

[row,col] = find(abs(mdl.Diagnostics.Dfbetas)>3/sqrt(100)); disp([row col])

2 1 28 1 84 1 93 1 2 2 13 3 84 3 2 4 84 4

`Dffits`

)The delete-1 scaled change in fitted values (`Dffits`

) show
the influence of each observation on the fitted response values.
`Dffits`

values with an absolute value larger than
2*sqrt(*p*/*n*) might be influential.

`Dffits`

for observation *i* is

$${\text{Dffits}}_{i}=s{r}_{i}\sqrt{\frac{{h}_{ii}}{1-{h}_{ii}}},$$

where *sr*_{i
}is the studentized residual, and
*h*_{ii} is the
leverage value of the fitted `LinearModel`

object.
`Dffits`

is an *n*-by-1 column vector in
the `Diagnostics`

table of the fitted
`LinearModel`

object. Each element in
`Dffits`

is the change in the fitted value caused by
deleting the corresponding observation and scaling by the standard error.

After obtaining a fitted model, say, `mdl`

, using
`fitlm`

or `stepwiselm`

, you can:

Display the

`Dffits`

values by indexing into the property using dot notationmdl.Diagnostics.Dffits

Plot the delete-1 scaled change in fitted values using

For details, see theplotDiagnostics(mdl,'Dffits')

`plotDiagnostics`

method of the`LinearModel`

class for details.

`Dffits`

This example shows how to determine the observations that are influential on the fitted response values using `Dffits`

values. Load the sample data and define the response and independent variables.

```
load hospital
y = hospital.BloodPressure(:,1);
X = double(hospital(:,2:5));
```

Fit a linear regression model.

mdl = fitlm(X,y);

Plot the `Dffits`

values.

`plotDiagnostics(mdl,'Dffits')`

The influential threshold limit for the absolute value of `Dffits`

in this example is 2*sqrt(5/100) = 0.45. Again, there are some observations with `Dffits`

values beyond the recommended limits.

Find the `Dffits`

values that are large in absolute value.

find(abs(mdl.Diagnostics.Dffits)>2*sqrt(4/100))

`ans = `*10×1*
2
13
28
44
58
70
71
84
93
95

`S2_i`

)The delete-1 variance (`S2_i`

) shows how the mean squared
error changes when an observation is removed from the data set. You can compare
the `S2_i`

values with the value of the mean squared
error.

`S2_i`

is a set of residual variance estimates obtained by
deleting each observation in turn. The `S2_i`

value for
observation *i* is

$$S2\_i=MS{E}_{\left(i\right)}=\frac{{\displaystyle \sum _{j\ne i}^{n}{\left[{y}_{j}-{\widehat{y}}_{j\left(i\right)}\right]}^{2}}}{n-p-1},$$

where
*y*_{j} is the
*j*th observed response value. `S2_i`

is
an *n*-by-1 vector in the `Diagnostics`

table
of the fitted `LinearModel`

object. Each element in
`S2_i`

is the mean squared error of the regression obtained
by deleting that observation.

After obtaining a fitted model, say, `mdl`

, using
`fitlm`

or `stepwiselm`

, you can:

Display the

`S2_i`

vector by indexing into the property using dot notationmdl.Diagnostics.S2_i

Plot the delete-1 variance values using

For details, see theplotDiagnostics(mdl,'S2_i')

`plotDiagnostics`

method of the`LinearModel`

class.

This example shows how to compute and plot S2_i values to examine the change in the mean squared error when an observation is removed from the data. Load the sample data and define the response and independent variables.

```
load hospital
y = hospital.BloodPressure(:,1);
X = double(hospital(:,2:5));
```

Fit a linear regression model.

mdl = fitlm(X,y);

Display the MSE value for the model.

mdl.MSE

ans = 23.1140

Plot the S2_i values.

`plotDiagnostics(mdl,'S2_i')`

This plot makes it easy to compare the S2_i values to the MSE value of 23.114, indicated by the horizontal dashed lines. You can see how deleting one observation changes the error variance.

`fitlm`

| `LinearModel`

| `plotDiagnostics`

| `plotResiduals`

| `stepwiselm`