Stratified Sampling

author: Dolores Ojeda, Gener Avilés R date: 2017-02-27 autosize: true

What is Stratified Sampling?

Population is partitioned in non-overlaping groups, called strata and a sample is collected from each stratum following a determined design.

Why use Stratified Sampling?

- May produce smaller error when estimating than simple random sample. Specially when measurements within strata have realitve small variation. - Cost by observation reduced. - There may be a need to have a subgroup (stratum) with similar estimates of those of the population.

Example

The Titanic Database:

     <U+FEFF>pclass            survived         name               sex           
 Min.   :1.000      Min.   :0.000   Length:1310        Length:1310       
 1st Qu.:2.000      1st Qu.:0.000   Class :character   Class :character  
 Median :3.000      Median :0.000   Mode  :character   Mode  :character  
 Mean   :2.295      Mean   :0.382                                        
 3rd Qu.:3.000      3rd Qu.:1.000                                        
 Max.   :3.000      Max.   :1.000                                        
 NA's   :1          NA's   :1                                            
      age              sibsp            parch          ticket         
 Min.   : 0.1667   Min.   :0.0000   Min.   :0.000   Length:1310       
 1st Qu.:21.0000   1st Qu.:0.0000   1st Qu.:0.000   Class :character  
 Median :28.0000   Median :0.0000   Median :0.000   Mode  :character  
 Mean   :29.8811   Mean   :0.4989   Mean   :0.385                     
 3rd Qu.:39.0000   3rd Qu.:1.0000   3rd Qu.:0.000                     
 Max.   :80.0000   Max.   :8.0000   Max.   :9.000                     
 NA's   :264       NA's   :1        NA's   :1                         
      fare            cabin             embarked        
 Min.   :  0.000   Length:1310        Length:1310       
 1st Qu.:  7.896   Class :character   Class :character  
 Median : 14.454   Mode  :character   Mode  :character  
 Mean   : 33.295                                        
 3rd Qu.: 31.275                                        
 Max.   :512.329                                        
 NA's   :2                                              
     boat                body        home.dest        
 Length:1310        Min.   :  1.0   Length:1310       
 Class :character   1st Qu.: 72.0   Class :character  
 Mode  :character   Median :155.0   Mode  :character  
                    Mean   :160.8                     
                    3rd Qu.:256.0                     
                    Max.   :328.0                     
                    NA's   :1189                      

Variable Codes

- Pclass: 1 = Upper, 2 = Middle, 3 = Lower. - SibSp: Number of Siblings/Spouses aboard. - Parch: Number of Parents/Children Aboard. - Embarked: C = Cherbourg, Q = Queenstown, S = Southampton.

Calculating Probabilities to select people who embarked in Queenstown

\(P(A) = \frac{\text{Numero de elementos de A}}{n}\)

There are 1310 entries, and 123 of them embarked in Queenstown, nevertheless the risk of dying was equally present for them as for the passengers from Southampton or Cherbourg.
If a uniform proability is calculated the numbers are:
- \(P(Q) = \frac{123}{1310} =\) 0.0938931 - \(P(C) = \frac{270}{1310} =\) 0.2061069 - \(P(S)\frac{914}{1310} =\) 0.6977099

This approximation will hinder the process of data mining and, eventually, the generation of a machine learning model for survival prediction.

Fixing the Problem

By using stratified sampling we can raise the probability for the group that boarded in Queenstown and survived to be selected, therefore, taken in consideration for the generation of a survival prediction model. For this we will use conditional probability:

\(P(Survived|EmbarkedQ)= \frac{P(Survived\cap EmbarkedQ)}{P(EmbarkedQ)} = \frac{44}{123}=\) 0.3577236

library(dplyr)
Q<-filter(titanic, embarked == "Q" & survived == 1)
count(Q)

# A tibble: 1 × 1
      n
  <int>
1    44

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