`ProSGPV`

is a package that performs variable selection with Second-Generation P-Values (SGPV). This document illustrates how `ProSGPV`

works with continuous outcomes in linear regression. Technical details about this algorithm can be found at Zuo, Stewart, and Blume (2020).

To install the `ProSGPV`

pacKakge from CRAN, you can do

`install("ProSGPV")`

Alternatively, you can install a development version of `ProSGPV`

by doing

`::install_github("zuoyi93/ProSGPV") devtools`

Once the package is installed, we can load the package to the current environment.

`library(ProSGPV)`

The data set is stored in the `ProSGPV`

package with the name `t.housing`

. The goal is to find important variables associated with the sale prices of real estate units and then build a prediction model. More details about data collection are available in Rafiei and Adeli (2016). There are 26 explanatory variables and one outcome, and variable description is shown below.

Category | Label | Description |
---|---|---|

Outcome | V9 | Actual sales price |

Project physical and financial features | V2 V3 V4 V5 V6 V7 V8 |
Total floor area of the building Lot area Total preliminary estimated construction cost Preliminary estimated construction cost Equivalent preliminary estimated construction cost in a selected base year Duration of construction Price of the unit at the beginning of the project |

Economic variables and indices | V11 V12 V13 V14 V15 V16 V17 V18 V19 V20 V21 V22 V23 V24 V25 V26 V27 V28 V29 |
The number of building permits issued Building services index for a pre-selected base year Wholesale price index of building materials for the base year Total floor areas of building permits issued by the city/municipality Cumulative liquidity Private sector investment in new buildings Land price index for the base year The number of loans extended by banks in a time resolution The amount of loans extended by banks in a time resolution The interest rate for loan in a time resolution The average construction cost by private sector when completed The average cost of buildings by private sector at the beginning Official exchange rate with respect to dollars Nonofficial (street market) exchange rate with respect to dollars Consumer price index (CPI) in the base year CPI of housing, water, fuel & power in the base year Stock market index Population of the city Gold price per ounce |

We can load the data and feed into `pro.sgpv`

function. By default, a two-stage algorithm is run and prints the indices of the selected variables.

```
<- t.housing[, -ncol(t.housing)]
x <- t.housing$V9
y
.2s <- pro.sgpv(x,y)
sgpv.2s
sgpv#> Selected variables are V8 V12 V13 V15 V17 V26
```

We can print the summary of the linear regression with selected variables with the S3 method `summary`

.

```
summary(sgpv.2s)
#>
#> Call:
#> lm(formula = Response ~ ., data = data.d)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -1276.35 -75.59 -9.58 59.46 1426.22
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 1.708e+02 3.471e+01 4.920 1.31e-06 ***
#> V8 1.211e+00 1.326e-02 91.277 < 2e-16 ***
#> V12 -2.737e+01 2.470e+00 -11.079 < 2e-16 ***
#> V13 2.185e+01 2.105e+00 10.381 < 2e-16 ***
#> V15 2.041e-03 1.484e-04 13.756 < 2e-16 ***
#> V17 -3.459e+00 8.795e-01 -3.934 0.00010 ***
#> V26 -4.683e+00 1.780e+00 -2.630 0.00889 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 194.8 on 365 degrees of freedom
#> Multiple R-squared: 0.9743, Adjusted R-squared: 0.9739
#> F-statistic: 2310 on 6 and 365 DF, p-value: < 2.2e-16
```

Coefficient estimates can be extracted by use of S3 method `coef`

. Note that it returns a vector of length \(p\).

```
coef(sgpv.2s)
#> [1] 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000
#> [6] 0.000000000 1.210755031 0.000000000 -27.367601037 21.853920174
#> [11] 0.000000000 0.002040784 0.000000000 -3.459496972 0.000000000
#> [16] 0.000000000 0.000000000 0.000000000 0.000000000 0.000000000
#> [21] 0.000000000 0.000000000 -4.683172725 0.000000000 0.000000000
#> [26] 0.000000000
```

In-sample prediction can be made using S3 method `predict`

and an external sample can be provided to make out-of-sample prediction with an argument of `newdata`

in the `predict`

function.

```
head(predict(sgpv.2s))
#> 1 2 3 4 5 6
#> 1565.7505 3573.7793 741.7576 212.1297 5966.1682 5724.0172
```

The `ProSGPV`

selection path can be extracted by use of S3 method `plot`

. `lambda.max`

argument controls the range of \(\lambda\). The black vertical dotted line is the \(\lambda\) selected by generalized information criterion (Fan and Tang (2013)). The null zone is the grey shaded region near 0. The blue labels on the Y-axis are the selected variables.

`plot(sgpv.2s,lambda.max = 0.005)`

By default, three lines per variables are provided. You can also choose to view only one bound per variable by setting `lpv`

argument to 1, where the one bound is the confidence bound that is closer to 0.

`plot(sgpv.2s, lambda.max=0.005, lpv=1)`

One-stage algorithm is available when \(n>p\) but may have reduced support recovery rate and higher parameter estimation bias. Its advantage is its fast computation speed and its result being fixed for a given data set.

```
.1s <- pro.sgpv(x,y,stage=1)
sgpv.1s
sgpv#> Selected variables are V8 V12 V13 V15 V17 V25 V26
```

Note that the one-stage algorithm selects one more variable than the two-stage algorithm.

S3 methods `summary`

, `coef`

, `predict`

and `plot`

are available for the one-stage algorithm. Particularly, `plot(sgpv.1s)`

would presents the variable selection results in the full model. Point estimates and 95% confidence intervals are shown for each variable, and the null bounds are shown in green vertical bars. Selected variables are colored in blue.

`plot(sgpv.1s)`