#+TITLE: Quantitative Methods
#+PROPERTY: header-args:R :session acj :eval never-export
#+STARTUP: hideall inlineimages hideblocks
#+HTML_HEAD: <style>#content{max-width:1200px;} </style>

* Title slide                                                         :slide:
#+BEGIN_SRC emacs-lisp-slide
(org-show-animate '("Quantitative Methods, Part-II" "Introduction to Statistical Inference" "Vikas Rawal" "Prachi Bansal" "" "" ""))
#+END_SRC

* Sampling Distributions

** Sampling Distributions                                            :slide:

# #+RESULTS: sampling2
[[file:bsample2.png]]

#+NAME: sampling2
#+BEGIN_SRC R :results output graphics :exports results :file bsample2.png :width 2500 :height 1500  :res 300
    library(data.table)
    readRDS("plfsdata/plfsacjdata.rds")->worker
    worker$standardwage->worker$wage
    #read.table("~/ssercloud/acj2018/worker.csv",sep=",",header=T)->worker
    c(1:nrow(worker))->worker$SamplingFrameOrder
    worker[sex!=3,]->worker
    library(ggplot2)
      ggplot(worker,aes(wage))+geom_density(colour="black",size=1)+scale_y_continuous(limits=c(0,0.05))+scale_x_continuous(limits=c(0,600),breaks=c(0,mean(worker$wage),1000))->p
  #    p+facet_wrap(~sex)->p
      p+annotate("text",x=380,y=0.045,
               label=paste("Population mean = ",round(mean(worker$wage)),sep=""))->p
      p+annotate("text",x=400,y=0.042,
               label="Distribution of sample means:")->p
     p+theme_bw()->p
      p



    sample(1:nrow(worker),5, replace=FALSE)->a1
    worker[a1,]->s1
    mean(s1$wage)->t1
    for (i in c(1:9999)) {
        sample(1:nrow(worker),5, replace=FALSE)->a1
        worker[a1,]->s1
        c(t1,mean(s1$wage))->t1
    }

    data.frame(sno=c(1:10000),meancol=t1)->t1
    p+geom_density(data=t1,aes(meancol),colour="blue",size=1)-> p
    paste("Sample size 5: mean = ",
          round(mean(t1$meancol)),
          "; stdev = ",
          round(sd(t1$meancol)),sep="")->lab
    p+annotate("text",x=450,y=0.030,label=lab,colour="blue")->p
    p

    sample(1:nrow(worker),20, replace=FALSE)->a1
    worker[a1,]->s1
    mean(s1$wage)->t0
    for (i in c(1:9999)) {
        sample(1:nrow(worker),20, replace=FALSE)->a1
        worker[a1,]->s1
        c(t0,mean(s1$wage))->t0
    }

    data.frame(sno=c(1:10000),meancol=t0)->t0
    p+geom_density(data=t0,aes(meancol),colour="darkolivegreen",size=1)-> p
    paste("Sample size 20: mean = ",
          round(mean(t0$meancol)),
          "; stdev = ",
          round(sd(t0$meancol)),sep="")->lab
    p+annotate("text",x=450,y=0.033,label=lab,colour="darkolivegreen")->p
    p

    sample(1:nrow(worker),50, replace=FALSE)->a1
    worker[a1,]->s1
    mean(s1$wage)->t
    for (i in c(1:9999)) {
        sample(1:nrow(worker),50, replace=FALSE)->a1
        worker[a1,]->s1
        c(t,mean(s1$wage))->t
    }

    data.frame(sno=c(1:10000),meancol=t)->t
    p+geom_density(data=t,aes(meancol),colour="red",size=1)-> p
    paste("Sample size 50: mean = ",
          round(mean(t$meancol)),
          "; stdev = ",
          round(sd(t$meancol)),sep="")->lab
    p+annotate("text",x=450,y=0.036,label=lab,colour="red")->p
    p

    sample(1:nrow(worker),200, replace=FALSE)->a1
    worker[a1,]->s1
    mean(s1$wage)->t4
    for (i in c(1:9999)) {
      sample(1:nrow(worker),200, replace=FALSE)->a1
      worker[a1,]->s1
      c(t4,mean(s1$wage))->t4
    }

    data.frame(sno=c(1:10000),meancol=t4)->t4
    p+geom_density(data=t4,aes(meancol),colour="pink",size=1)-> p
    paste("Sample size 200: mean = ",
          round(mean(t4$meancol)),
          "; stdev = ",
          round(sd(t4$meancol)),sep="")->lab
    p+annotate("text",x=450,y=0.039,label=lab,colour="pink")->p
    p
#+end_src

** Sampling Distributions                                            :slide:

+ $Standard.error = \frac{\sigma}{\sqrt{mean}}$


|                                    Variable | Value |
|---------------------------------------------+-------|
| Standard deviation of population ($\sigma$) |   130 |
|          Standard errors of samples of size |       |
|                                           5 |    58 |
|                                          20 |    29 |
|                                          50 |    18 |
|                                         200 |     9 |




* Introduction to Hypothesis Testing
** Transforming the Distribution to Standard Normal                  :slide:

#+RESULTS: sampling3
[[file:bsample3.png]]

#+NAME: sampling3
#+BEGIN_SRC R :results output graphics :exports results :file bsample3.png :width 2500 :height 2000  :res 300
  library(data.table)
  readRDS("plfsdata/plfsacjdata.rds")->worker
  worker$standardwage->worker$wage
  c(1:nrow(worker))->worker$SamplingFrameOrder
  worker[sex!=3,]->worker
  library(ggplot2)

  worker->t9
  (t9$wage-mean(t9$wage))/sd(t9$wage)->t9$wage
  ggplot(t9,aes(wage))+geom_density(colour="black",size=1)->p
  p+scale_y_continuous(limits=c(0,0.75))->p
  p+scale_x_continuous(limits=c(-15,15)
                      ,breaks=c(-5,0,mean(worker$wage),10,15))->p
  p+theme_bw()->p
  p



  sample(1:nrow(worker),5, replace=FALSE)->a1
  worker[a1,]->s1
  mean(s1$wage)->t1
    for (i in c(1:9999)) {
        sample(1:nrow(worker),5, replace=FALSE)->a1
        worker[a1,]->s1
        c(t1,mean(s1$wage))->t1
    }

  data.frame(sno=c(1:10000),meancol=(t1-mean(worker$wage))/sd(t1))->t1
  p+geom_density(data=t1,aes(meancol),colour="blue",size=1)-> p
  p

  sample(1:nrow(worker),20, replace=FALSE)->a1
  worker[a1,]->s1
  mean(s1$wage)->t0
  for (i in c(1:9999)) {
    sample(1:nrow(worker),20, replace=FALSE)->a1
    worker[a1,]->s1
    c(t0,mean(s1$wage))->t0
  }

  data.frame(sno=c(1:10000),meancol=(t0-mean(worker$wage))/sd(t0))->t0
  p+geom_density(data=t0,aes(meancol),colour="darkolivegreen",size=1)-> p
  p

  sample(1:nrow(worker),50, replace=FALSE)->a1
  worker[a1,]->s1
  mean(s1$wage)->t
  for (i in c(1:9999)) {
    sample(1:nrow(worker),50, replace=FALSE)->a1
    worker[a1,]->s1
    c(t,mean(s1$wage))->t
  }

  data.frame(sno=c(1:10000),meancol=(t-mean(worker$wage))/sd(t))->t
  p+geom_density(data=t,aes(meancol),colour="red",size=1)-> p
  p

  sample(1:nrow(worker),200, replace=FALSE)->a1
  worker[a1,]->s1
  mean(s1$wage)->t4
  for (i in c(1:9999)) {
    sample(1:nrow(worker),200, replace=FALSE)->a1
    worker[a1,]->s1
    c(t4,mean(s1$wage))->t4
  }

  data.frame(sno=c(1:10000),meancol=(t4-mean(worker$wage))/sd(t4))->t4
  p+geom_density(data=t4,aes(meancol),colour="pink",size=1)-> p
  p
#+end_src


** Distribution of sample mean with unknown population variance      :slide:

#+RESULTS: sampling5
[[file:bsample5.png]]

#+NAME: sampling5
#+BEGIN_SRC R :results output graphics :exports results :file bsample5.png :width 3500 :height 2000  :res 300
  library(data.table)
  library(ggplot2)
  options(scipen=9999)
  readRDS("plfsdata/plfsacjdata.rds")->worker
  worker$standardwage->worker$wage
  c(1:nrow(worker))->worker$SamplingFrameOrder
  worker[sex!=3,]->worker

  worker->t9
  (t9$wage-mean(t9$wage))/sd(t9$wage)->t9$wage
  ggplot(t9,aes(wage))+geom_density(colour="black",size=1)->p
  p+scale_y_continuous(limits=c(0,0.75))->p
  p+scale_x_continuous(limits=c(-15,15)
                      ,breaks=c(-15,0,round(mean(worker$wage)),15))->p
  p+theme_bw()->p
  p

  data.frame(sno=c(),meancol=c(),sterr=c())->t4
  samplesize=10
  for (i in c(1:20000)) {
    sample(1:nrow(worker),samplesize, replace=FALSE)->a1
    worker[a1,]->s1
    rbind(t4,data.frame(
               sno=i,
               meancol=mean(s1$wage),
               sterr=sd(s1$wage)/sqrt(samplesize)
             )
          )->t4
  }

  (t4$meancol)/t4$sterr->t4$teststat
  (t4$meancol)/sd(t4$meancol)->t4$teststat2
  data.frame(modelt=rt(200000,samplesize-1,ncp=mean(t4$teststat)),modelnorm=rnorm(200000,mean=mean(t4$teststat2)))->m

  sd(t4$teststat)
  sd(m$modelt)
  sd(m$modelnorm)
  sd(t4$teststat2)
  mean(t4$teststat)
  mean(m$modelt)
  mean(m$modelnorm)
  mean(t4$teststat2)

  ggplot()->p
  p+geom_density(data=t4,aes(teststat2),colour="red",size=1)-> p
  p+geom_density(data=m,aes(modelnorm),colour="black",size=1)->p
  p+geom_density(data=t4,aes(teststat),colour="blue",size=1)-> p
  p+geom_density(data=m,aes(modelt),colour="darkolivegreen",size=1)->p
  p+annotate("text",x=-30,y=0.42,
             label=paste("Normal distribution, with standard deviation",round(sd(m$modelnorm),2)),
             colour="black",hjust=0)->p
  p+annotate("text",x=-30,y=0.40,
             label=paste("Statistic with known population variance, standard error =",
                         round(sd(t4$teststat2),2)),
             colour="red",hjust=0)->p
  p+annotate("text",x=-30,y=0.38,
             label=paste("t distribution, with standard deviation =",round(sd(m$modelt),2)),
             colour="darkolivegreen",hjust=0)->p
  p+annotate("text",x=-30,y=0.36,
             label=paste("Statistic with unknown population variance, standard error =",
                         round(sd(t4$teststat),2)),
             colour="blue",hjust=0)->p
  p+scale_x_continuous(limits=c(-30,30))+theme_bw()->p
  p
#+end_src




** t-test for means                                                  :slide:

***  Testing if the mean is different from a specified value (say zero) :slide:

#+name: ttest1
#+begin_src R :results output list org
  readRDS("plfsdata/plfsacjdata.rds")->worker
  worker$standardwage->worker$wage
  worker->t9
  t.test(t9$wage)
#+end_src

#+RESULTS: ttest1
#+begin_src org
- One Sample t-test
- data:  t9$wage
- t = 432.99, df = 37634, p-value < 0.00000000000000022
- alternative hypothesis: true mean is not equal to 0
- 95 percent confidence interval:
- 289.7136 292.3484
- sample estimates:
- mean of x 
- 291.031
#+end_src


*** Testing equality of means                                       :slide:

+ Here we test if the mean wages of men and women are equal.

#+name: ttest2
#+begin_src R :results output list org
  subset(worker,sex!=3)->t9
  factor(t9$sex)->t9$sex
  t.test(wage~sex,data=t9)
#+end_src

#+RESULTS: ttest2
#+begin_src org
- Welch Two Sample t-test
- data:  wage by sex
- t = 79.02, df = 13483, p-value < 0.00000000000000022
- alternative hypothesis: true difference in means is not equal to 0
- 95 percent confidence interval:
- 104.6563 109.9805
- sample estimates:
- mean in group 1 mean in group 2 
- 310.8974        203.5790
#+end_src


** Testing for equality of proportions                               :slide:

+ Here we test if proportion of person who have passed high school is different for men and women

#+name: proptest1
#+begin_src R :results output list org
  subset(worker,sex!=3)->t9
  as.numeric(t9$gen_edu_level)->t9$gen_edu_level
  factor(t9$sex)->t9$sex
  t9[gen_edu_level>=8,.(schooled=length(fsu)),.(sex)]->a
  t9[,.(all=length(fsu)),.(sex)]->b
  prop.test(a$schooled,b$all)
#+end_src


#+CAPTION: Results of test for equality of proportions of men and women who have passed secondary school
#+RESULTS: proptest1
#+begin_src org
- 2-sample test for equality of proportions with continuity correction
- data:  a$schooled out of b$all
- X-squared = 847.73, df = 1, p-value < 0.00000000000000022
- alternative hypothesis: two.sided
- 95 percent confidence interval:
- -0.1694726 -0.1525728
- sample estimates:
- prop 1     prop 2 
- 0.09245986 0.25348253
#+end_src