Elemental Descriptive and Inferential Statistics (LearningStats)

Description

This package provides tools to teach students elemental Statistics. The main topics covered are Descriptive Statistics, Probability models (discrete and continuous variables) and Statistical Inference (confidence intervals and hypothesis tests).

Details

Main sections of LearningStats-package are:

A.- Data

This section includes a function to read different file extensions and a dataset on health-related behaviours with 18 variables. The main advantage of this tool is that with just one single function most of the common file extensions can be imported into R.

B.- Descriptive Statistics

The functions included in this section perform Descriptive Statistics by quantitatively describing or summarizing different characteristics from a sample. Graphical tools are also available.

C.- Probability models

In this section probability models for discrete and continuous variables are provided.

C.1-Discrete variables:

The user is allowed to display, with several options, the probability mass and/or distribution function for the following discrete distributions: Binomial, Discrete Uniform, Hypergeometric, Negative Binomial and Poisson.

C.2-Continuous variables:

The user is allowed to display, with several options, the density, distribution and/or quantile functions for the following continuous distributions: Beta, Chi-squared, Exponential, F-Snedecor, Gamma, Normal, T-Student and Uniform.

C.3-Illustrations:

Also in this section three common approximations between different distributions are illustrated. The approximations considered are: the Normal approximation to Binomial, the Normal approximation to Poisson and the Poisson approximation to Binomial.

D.- Statistical Inference

This section includes functions to perform Statistical Inference (confidence intervals and hypothesis testing) with one or two populations and also for categorical data.

D.1-Confidence intervals:

The functions included here provide pointwise and confidence interval estimation for different population parameters. One or two populations are supported.

One population:

Two populations:

D.2-Hypothesis tests:

This sections allows to compute hypothesis tests for different population parameters (mean, variance and proportion) in one or two populations. The scenarios covered here are those mentioned in the Confidence Interval section as well as a Chi-squared independence test.

One population:

Two populations:

Categorical data:

E.- Regression

This section includes a function to describe the relationship between two continuous variables through a simple linear regression model, providing the R-squared coefficient.

Illustration of the Normal approximation to Binomial

Description

When certain conditions are met (see Details), the Binomial distribution can be approximated by the Normal one. The function AproxBinomNorm illustrates this fact by plotting the mass diagram corresponding with the discrete distribution (parameters are given by the user) on which the associated Normal density function is also displayed.

Usage

AproxBinomNorm(n, p, legend = TRUE, xlab = "", ylab = "Probability",
  main = "Normal approximation to Binomial", col.fill = "grey",
  col.line = "red", lwd = 2)

Arguments

Details

The approximation is accurate only if one of these three conditions is met:

- p in (0.1, 0.9) and n>=30,

- p in [0,0.1] and np>5,

- p in [0.9,1] and n(1-p)>5.

Value

This function is called for the side effect of drawing the plot.

Examples

n=45; p=0.4
AproxBinomNorm(n,p)
AproxBinomNorm(n,p,col.fill="blue",col.line="orange")
AproxBinomNorm(n,p,legend=FALSE)

Illustration of the Poisson approximation to Binomial

Description

AproxBinomPois represents the probability mass associated with a Binomial distribution with certain parameters n and p joint with the Poisson distribution with mean equal to np. Note that the Binomial distribution can be approximated by a Poisson distribution when certain conditions are met (see Details).

Usage

AproxBinomPois(n, p, xlab = "x", ylab = "Probability Mass",
  main = "Poisson approximation to Binomial distribution", col1 = "grey",
  col2 = "red")

Arguments

Details

The approximation is accurate only if one of these conditions is met:

- p in (0,0.1), n>=30 and np<5,

- p in (0.9,1), n>=30 and n(1-p)<5. Note that given X1 a Binomial distribution with parameters n and p, and X2 a Binomial distribution with parameters n and 1-p, it follows that P(X1=a)=P(X2=n-a). Then, the variable X2 can be approximated to a Poisson distribution with parameter lambda=n(1-p) and this Poisson distribution can be used in order to approximate the mass probability function associated with X1.

Value

This function is called for the side effect of drawing the plot.

Examples

n=50;p=0.93
AproxBinomPois(n,p)
n=100;p=0.03
AproxBinomPois(n,p)

Illustration of the Normal approximation to Poisson

Description

When certain conditions are met (see Details), the Poisson distribution can be approximated by the Normal one. The function AproxPoisNorm illustrates this fact by plotting the mass diagram corresponding with the discrete distribution (parameter is given by the user) on which the associated Normal density function is also displayed.

Usage

AproxPoisNorm(lambda, legend = TRUE, xlab = "", ylab = "Probability",
  main = "Normal approximation to Poisson", col.fill = "grey",
  col.line = "red", lwd = 2)

Arguments

Details

The approximation is accurate only if lambda>=10.

Value

This function is called for the side effect of drawing the plot.

Examples

lambda=15
AproxPoisNorm(lambda)
AproxPoisNorm(lambda,col.fill="blue",col.line="orange")
AproxPoisNorm(lambda,legend=FALSE)

Boxplot Representation

Description

The function BoxPlot displays a boxplot representation of a given sample.

Usage

BoxPlot(x, col = "white", main = "Boxplot representacion", ylab = "",
  legend = TRUE)

Arguments

Details

The quantiles needed to obtain this representation are computed using the function sample.quantile.

Value

This function is called for the side effect of drawing the plot.

Examples

x=c(5,-5,rnorm(40))
BoxPlot(x,col="pink")

Plot a Histogram

Description

The function Histogram plots a histogram of a given sample.

Usage

Histogram(x, freq = FALSE, col.fill = "grey", main = "autom",
  xlab = "", ylab = "autoy")

Arguments

Details

The procedure to construct the histogram is detailed below:

- number of intervals: the closest integer to sqrt(n);

- amplitude of each interval: the range of the sample divided by the number of intervals, i.e., the breaks are equidistant and rounded to two decimals;

- height of each bar: by default (freq=FALSE) the plotted histogram is a density (area 1); if freq=TRUE, then the values of the bars are the absolute frequencies.

Value

A list containing the following components:

Independently on the user saving those values, the function provides the frequency table on the console.

Examples

x=rnorm(10)
Histogram(x)
Histogram(x,freq=TRUE)
Histogram(x,freq=TRUE,col="pink")

Confidence Interval for the Mean of a Normal Population

Description

Mean.CI provides a pointwise estimation and a confidence interval for the mean of a Normal population in both scenarios: known and unknown population variance.

Usage

Mean.CI(x, sigma = NULL, sc = NULL, s = NULL, n = NULL, conf.level)

Arguments

Details

The formula interface is applicable when the user provides the sample and also when the user provides the value of the sample characteristics (sample mean, cuasi-standard deviation or sample standard deviation, jointly with the sample size).

Value

A list containing the following components:

Independently on the user saving those values, the function provides a summary of the result on the console.

Examples

#Given the sample with known population variance
dat=rnorm(20,mean=2,sd=1)
Mean.CI(dat, sigma=1, conf.level=0.95)

#Given the sample with unknown population variance
dat=rnorm(20,mean=2,sd=1)
Mean.CI(dat, conf.level=0.95)

#Given the sample mean with known population variance:
dat=rnorm(20,mean=2,sd=1)
Mean.CI(mean(dat),sigma=1,n=20,conf.level=0.95)

#Given the sample mean with unknown population variance:
dat=rnorm(20,mean=2,sd=1)
Mean.CI(mean(dat),sc=sd(dat),n=20,conf.level=0.95)

One Sample Mean Test of a Normal Population

Description

Mean.test allows to compute hypothesis tests for a Normal population mean in both scenarios: known and unknown population variance.

Usage

Mean.test(x, mu0, sigma = NULL, sc = NULL, s = NULL, n = NULL,
  alternative = "two.sided", alpha = 0.05, plot = TRUE, lwd = 1)

Arguments

Details

The formula interface is applicable when the user provides the sample and also when the user provides the value of the sample characteristics (sample mean, cuasi-standard deviation or sample standard deviation, jointly with the sample size).

Value

A list with class "lstest" and "htest" containing the following components:

Examples

x <- rnorm(50, mean = 4, sd = 2)
#unknown sigma
Mean.test(x, mu0 = 3.5)
Mean.test(mean(x), sc = sd(x),  n = length(x), mu0 = 3.5)
#known sigma
Mean.test(x, mu0 = 3.5, sigma = 2)

Variance Estimator when the Population Mean is Known

Description

S2mu computes a estimation of the variance, given a sample x with known population mean (denoted by mu).

Usage

S2mu(x, mu)

Arguments

Details

Given \{x_1,\ldots,x_n\} a sample of a random variable, the variance estimator when the population mean (denoted by \mu) is known can be computed as S^2_\mu=\frac{1}{n}\sum_{i=1}^n (x_i-\mu)^2.

Value

A single numerical value corresponding with the variance estimation when the population mean is known.

Examples

x=rnorm(20)
S2mu(x,mu=0)

Standard Deviation Estimator when the Population Mean is Known

Description

Smu computes a estimation of the standard deviation, given a sample x with known mean (denoted by mu).

Usage

Smu(x, mu)

Arguments

Details

Given \{x_1,\ldots,x_n\} a sample of a random variable, the standard deviation estimator when the population mean (denoted by \mu) is known can be computed as S_\mu=\sqrt{\frac{1}{n}\sum_{i=1}^n (x_i-\mu)^2}.

Value

A single numerical value corresponding with the standard deviation estimation when the population mean is known.

Examples

x=rnorm(20)
Smu(x,mu=0)

Confidence Interval for the Difference between the Means of Two Normal Populations.

Description

diffmean.CI provides a pointwise estimation and a confidence interval for the difference between the means of two Normal populations in different scenarios: population variances known or unknown, population variances assumed equal or not, and paired or independent populations.

Usage

diffmean.CI(x1, x2, sigma1 = NULL, sigma2 = NULL, sc1 = NULL,
  sc2 = NULL, s1 = NULL, s2 = NULL, n1 = NULL, n2 = NULL,
  paired = FALSE, var.equal = FALSE, conf.level)

Arguments

Details

If sigma1 and sigma2 are given, known population variances formula is applied; the unknown one is used in other case.

If paired is TRUE then both x1 and x2 must be specified and their sample sizes must be the same. If paired is null, then it is assumed to be FALSE.

For var.equal=TRUE, the formula of the pooled variance is \frac{(n1-1)sc1^2+(n2-1)sc2^2}{n1+n2-2}.

Value

A list containing the following components:

Independently on the user saving those values, the function provides a summary of the result on the console.

Examples

#Given unpaired samples with known population variance
dat1=rnorm(20,mean=2,sd=1);dat2=rnorm(30,mean=2,sd=1.5)
diffmean.CI(dat1,dat2,sigma1=1,sigma2=1.5,conf.level=0.9)

#Given unpaired samples with unknown but equal population variances
dat1=rnorm(20,mean=2,sd=1);dat2=rnorm(30,mean=2,sd=1)
diffmean.CI(dat1,dat2,paired=FALSE,var.equal=TRUE,conf.level=0.9)

#Given the characteristics of unpaired samples with unknown and different population variances
dat1=rnorm(20,mean=2,sd=1);dat2=rnorm(30,mean=2,sd=1)
x1=mean(dat1);x2=mean(dat2);sc1=sd(dat1);sc2=sd(dat2);n1=length(dat1);n2=length(dat2)
diffmean.CI(x1,x2,sc1=sc1,sc2=sc2,n1=n1,n2=n2,paired=FALSE,var.equal=FALSE,conf.level=0.9)

#Given paired samples
dat1=rnorm(20,mean=2,sd=1);dat2=dat1+rnorm(20,mean=0,sd=0.5)
diffmean.CI(dat1,dat2,paired=TRUE,conf.level=0.9)

Two Sample Mean Test of Normal Populations

Description

diffmean.test allows to compute hypothesis tests about two population means. The difference between the means of two Normal populations is tested in different scenarios: known or unknown variance, variances assumed equal or different and paired or independent populations.

Usage

diffmean.test(x1, x2, sigma1 = NULL, sigma2 = NULL, sc1 = NULL,
  sc2 = NULL, s1 = NULL, s2 = NULL, n1 = NULL, n2 = NULL,
  var.equal = FALSE, paired = FALSE, alternative = "two.sided",
  alpha = 0.05, plot = TRUE, lwd = 1)

Arguments

Details

If sigma1 and sigma2 are given, known population variances formula is applied; the unknown one is used in other case.

If paired is TRUE then both x1 and x2 must be specified and their sample sizes must be the same. If paired is null, then it is assumed to be FALSE.

For var.equal=TRUE, the formula of the pooled variance is \frac{(n1-1)sc1^2+(n2-1)sc2^2}{n1+n2-2}.

Value

A list with class "lstest" and "htest" containing the following components:

Examples

x1 <- rnorm(40, mean = 1.5, sd = 2)
x2 <- rnorm(60, mean = 2, sd = 2)
#equal variances
diffmean.test(x1, x2, var.equal = TRUE)
diffmean.test(mean(x1), mean(x2),
              sc1 = sd(x1), sc2 = sd(x2),
              n1 = 40, n2 = 60, var.equal = TRUE)
x3 <- rnorm(60, mean = 2, sd = 1.5)
#different variances
diffmean.test(x1, x3)
#known standard deviation
diffmean.test(x1, x3, sigma1 = 2, sigma2 = 1.5)
x4 <- x1 + rnorm(40, mean = 0, sd = 0.1)
#paired samples
diffmean.test(x1, x4, paired = TRUE)

Large Sample Confidence Interval for the Difference between Two Population Proportions

Description

diffproportion.CI provides a pointwise estimation and a confidence interval for the difference between two population proportions.

Usage

diffproportion.CI(x1, x2, n1, n2, conf.level)

Arguments

Details

Counts of successes and failures must be nonnegative and hence not greater than the corresponding numbers of trials which must be positive. All finite counts should be integers. If the number of successes are given, then the proportion estimate is computed.

Value

A list containing the following components:

Independently on the user saving those values, the function provides a summary of the result on the console.

Examples

#Given the sample proportion estimate
diffproportion.CI(0.3,0.4,100,120,conf.level=0.95)

#Given the number of successes
diffproportion.CI(30,48,100,120,conf.level=0.95)

#Given in one sample the number of successes and in the other the proportion estimate
diffproportion.CI(0.3,48,100,120,conf.level=0.95)

Two Sample Proportion Test

Description

diffproportion.test allows to compute hypothesis tests about two population proportions.

Usage

diffproportion.test(x1, x2, n1, n2, alternative = "two.sided",
  alpha = 0.05, plot = TRUE, lwd = 1)

Arguments

Details

Counts of successes and failures must be nonnegative and hence not greater than the corresponding numbers of trials which must be positive. All finite counts should be integers. If the number of successes is given, then the proportion estimate is computed.

Value

A list with class "lstest" and "htest" containing the following components:

Examples

x1 <- rbinom(1, 120, 0.6)
x2 <- rbinom(1, 100, 0.6)
diffproportion.test(x1 = x1, x2 = x2, n1 = 120, n2 = 100)
diffproportion.test(x1 = 0.6, x2 = 0.65, n1 = 120, n2 = 100)

Confidence Interval for the Ratio Between the Variances of Two Normal Populations

Description

diffvariance.CI provides a pointwise estimation and a confidence interval for the ratio of Normal population variances in both scenarios: known and unknown population mean.

Usage

diffvariance.CI(x1 = NULL, x2 = NULL, s1 = NULL, s2 = NULL,
  sc1 = NULL, sc2 = NULL, smu1 = NULL, smu2 = NULL, mu1 = NULL,
  mu2 = NULL, n1 = NULL, n2 = NULL, conf.level)

Arguments

Details

The formula interface is applicable when the user provides the sample and when the user provides the value of the sample characteristics (sample mean, cuasi-standard deviation or sample standard deviation, and sample size). Moreover, when mu1, smu1, mu2 or smu2 are provided, the function performs the procedure with known population means, and unknown in other case.

Value

A list containing the following components:

Independently on the user saving those values, the function provides a summary of the result on the console.

Examples

#Given the samples with known population means
dat1=rnorm(20,mean=2,sd=1); dat2=rnorm(30,mean=3,sd=1)
diffvariance.CI(x1=dat1,x2=dat2,mu1=2,mu2=3,conf.level=0.95)

#Given the sample standard deviations with known population means
dat1=rnorm(20,mean=2,sd=1); dat2=rnorm(30,mean=3,sd=1)
smu1=Smu(dat1,mu=2);smu2=Smu(dat2,mu=3)
diffvariance.CI(smu1=smu1,smu2=smu2,n1=20,n2=30,conf.level=0.95)

#Given the samples with unknown population means
dat1=rnorm(20,mean=2,sd=1); dat2=rnorm(30,mean=3,sd=1)
diffvariance.CI(x1=dat1,x2=dat2,conf.level=0.95)

#Given the sample standard deviations with unknown population means
dat1=rnorm(20,mean=2,sd=1); dat2=rnorm(30,mean=3,sd=1)
diffvariance.CI(s1=(19/20)*sd(dat1),s2=(29/30)*sd(dat2),n1=20,n2=30,conf.level=0.95)

Two Sample Variance Test of Normal Populations

Description

diffvariance.test allows to compute hypothesis tests about two population variances in both scenarios: known and unknown population mean.

Usage

diffvariance.test(x1 = NULL, x2 = NULL, s1 = NULL, s2 = NULL,
  sc1 = NULL, sc2 = NULL, smu1 = NULL, smu2 = NULL, mu1 = NULL,
  mu2 = NULL, n1 = NULL, n2 = NULL, alternative = "two.sided",
  alpha = 0.05, plot = TRUE, lwd = 1)

Arguments

Details

The formula interface is applicable when the user provides the sample(s) or values of the sample characteristics (cuasi-standard deviation or sample standard deviation). When mu1 and mu2 or smu1 and smu2 are provided, the function performs the procedure with known population means.

Value

A list with class "lstest" and "htest" containing the following components:

Examples

x1 <- rnorm(40, mean = 1, sd = 2)
x2 <- rnorm(60, mean = 2, sd = 1.5)
# unknown population mean
diffvariance.test(x1, x2)
diffvariance.test(x1, sc2 = sd(x2), n2 = length(x2))
diffvariance.test(sc1 = sd(x1), sc2 = sd(x2), n1 = length(x1), n2 = length(x2))
# known population mean
diffvariance.test(x1, x2, mu1 = 1, mu2 = 2)
smu1 <- Smu(x1, mu = 1); smu2 <- Smu(x2, mu = 2)
diffvariance.test(smu1 = smu1, smu2 = smu2, n1 = length(x1), n2 = length(x2))

Plot a Cumulative Frequency Polygon

Description

The function freq.pol computes a cumulative frequency polygon of a given sample.

Usage

freq.pol(x, freq = FALSE, col = "black", lwd = 2, main = "autom",
  xlab = "", ylab = "autoy", bar = TRUE, fill = TRUE,
  col.fill = "pink")

Arguments

Details

The sample must be numeric and coming from a continuous variable.

The procedure used to define the intervals for the frequency table and the bars (if plotted) is the same as used for the histogram performed in this package (see ?Histogram).

Value

A list containing the following components:

Independently on the user saving those values, the function provides the frequency table on the console.

Examples

x=rnorm(10)
freq.pol(x)

freq.pol(x,freq=TRUE,fill=TRUE,col.fill="yellow")

Frequency Table

Description

The function freq.table computes a frequency table with absolute and relative frequencies (for non-ordered variables); and with those as well as their cumulative counterparts (for ordered variables).

Usage

freq.table(x, cont, ord = NULL)

Arguments

Details

The procedure used to define the intervals for the frequency table in the continuous case is the same as used for the histogram (see ?Histogram).

Value

A list containing the following components:

The values of the cumulative frequencies (Ni and Fi) are only computed and provided when the variable of interest is ordered. If the user does not save those values, the function provides the list on the console.

Examples

#Nominal variable
x=sample(c("yellow","red","blue","green"),size=20,replace=TRUE)
freq.table(x,cont=FALSE)

#Ordinal variable
x=sample(c("high","small","medium"),size=20,replace=TRUE)
freq.table(x,cont=FALSE,ord=c("small","medium","high"))

#Discrete variable
x=sample(1:5,size=20,replace=TRUE)
freq.table(x,cont=FALSE)

#Continuous variable
x=rnorm(20)
freq.table(x,cont=TRUE)

Chi-squared Independence Test for Categorical Data.

Description

indepchisq.test allows to computes Chi-squared independence hypothesis test for two categorical values.

Usage

indepchisq.test(Oij, x, y, alpha = 0.05, plot = TRUE, lwd = 1)

Arguments

Details

The expected frequencies are calculated as follows

E_{ij}=\frac{n_{i\bullet}\times n_{\bullet j}}{n},

and the test statistic is given by

T = \sum_{i,j} \frac{(n_{ij} - E_{ij})^2}{E_{ij}},

T \in \chi^2_{(r-1)(s-1)}, where n is the number of observations, n_{i\bullet} is the marginal frequency of category i of variable x, n_{\bullet j} is the marginal frequency of category j of variable y, r is the number of categories in variable x and s the number of categories in variable y.

The null hypothesis is rejected when T > \chi^2_{(r-1)(s-1),1-\alpha}, where \chi^2_{(r-1)(s-1),1-\alpha} is the 1-\alpha quantile of a \chi^2 distribution with (r-1)(s-1) degrees of freedom.

Value

A list with class "lstest" and "htest" containing the following components:

Examples

Oij <- matrix(c( 20,    8,
                934, 1070,
                113,   92), ncol = 2, byrow = TRUE)
indepchisq.test(Oij)

Density Function, Distribution Function and/or Quantile Function Representations associated with a Beta Distribution

Description

plotBeta represents density, distribution and/or quantile functions associated with a Beta distribution with parameters shape1 and shape2.

Usage

plotBeta(shape1, shape2, type = "b", col = "black")

Arguments

Value

This function is called for the side effect of drawing the plot.

Examples

shape1=1;shape2=1
plotBeta(shape1,shape2)
plotBeta(shape1,shape2,col="red")
plotBeta(shape1,shape2,type="q")
plotBeta(shape1,shape2,type="dis")
plotBeta(shape1,shape2,type="den")

Probability Mass and/or Distribution Function Representations associated with a Binomial Distribution

Description

plotBinom represents the probability mass and/or the distribution function associated with a Binomial distribution with certain parameters n and p.

Usage

plotBinom(n, p, type = "b", col = "grey")

Arguments

Details

Note that if n=1, the Binomial distribution is also known as Bernoulli distribution.

Value

A matrix containing the probability mass and the distribution function associated with each point of the support of a Binomial distribution with parameters n and p.

This function is called for the side effect of drawing the plot.

Examples

n=10;p=0.3
plotBinom(n,p,type="d")
plotBinom(n,p,type="p",col="pink")
plotBinom(n,p)

Density Function, Distribution Function and/or Quantile Function Representations associated with a Chi-squared Distribution

Description

plotChi represents density, distribution and/or quantile functions associated with a Chi-squared distribution with df degrees of freedom.

Usage

plotChi(df, type = "b", col = "black")

Arguments

Value

This function is called for the side effect of drawing the plot.

Examples

df=10
plotChi(df)
plotChi(df,col="red")
plotChi(df,type="q")
plotChi(df,type="dis")
plotChi(df,type="den")

Probability Mass and/or Distribution Function Representations associated with a Discrete Uniform Distribution

Description

plotDUnif represents the probability mass and/or the distribution function associated with a Discrete Uniform distribution with support x.

Usage

plotDUnif(x, type = "b", col = "grey")

Arguments

Value

A matrix containing the probability mass and the distribution function associated with each point of the support (denoted by x) of a Discrete Uniform distribution.

Examples

x=1:5
plotDUnif(x,type="d")
plotDUnif(x,type="p",col="pink")
plotDUnif(x)

Density Function, Distribution Function and/or Quantile Function Representations associated with a Exponential Distribution

Description

plotExp represents density, distribution and/or quantile functions associated with a Exponential distribution with certain parameter lambda.

Usage

plotExp(lambda, type = "b", col = "black")

Arguments

Value

This function is called for the side effect of drawing the plot.

Examples

lambda=0.5
plotExp(lambda)
plotExp(lambda,col="red")
plotExp(lambda,type="q")
plotExp(lambda,type="dis")
plotExp(lambda,type="den")

Density Function, Distribution Function and/or Quantile Function Representations associated with a F-Snedecor Distribution

Description

plotBeta represents density, distribution and/or quantile functions associated with a F-Snedecor distribution with certain df1 and df2 degrees of freedom.

Usage

plotFS(df1, df2, type = "b", col = "black")

Arguments

Value

This function is called for the side effect of drawing the plot.

Examples

df1=10;df2=15
plotFS(df1,df2)
plotFS(df1,df2,col="red")
plotFS(df1,df2,type="q")
plotFS(df1,df2,type="dis")
plotFS(df1,df2,type="den")

Density Function, Distribution Function and/or Quantile Function Representations associated with a Gamma Distribution

Description

plotGamma represents density, distribution and/or quantile functions associated with a Gamma distribution with certain parameters lambda and shape.

Usage

plotGamma(lambda, shape, type = "b", col = "black")

Arguments

Value

This function is called for the side effect of drawing the plot.

Examples

lambda=0.5;shape=4
plotGamma(lambda,shape)
plotGamma(lambda,shape,col="red")
plotGamma(lambda,shape,type="q")
plotGamma(lambda,shape,type="dis")
plotGamma(lambda,shape,type="den")

Probability Mass and/or Distribution Function Representations associated with a Hypergeometric Distribution

Description

plotHyper represents the probability mass and/or the distribution function associated with a Hypergeometric distribution with parameters N, n and k.

Usage

plotHyper(N, n, k, type = "b", col = "grey")

Arguments

Value

A matrix containing the probability mass and the distribution function associated with each point of the support of a Hypergeometric distribution with parameters N, n and k.

Examples

N=20;n=12;k=5
plotHyper(N,n,k,type="d")
plotHyper(N,n,k,type="p",col="pink")
plotHyper(N,n,k)

Probability Mass and/or Distribution Function Representations associated with a Negative Binomial Distribution

Description

plotNegBinom represents the probability mass and/or the distribution function associated with a Negative Binomial distribution with certain parameters n and p.

Usage

plotNegBinom(n, p, type = "b", col = "grey")

Arguments

Details

Note that if n=1, the Negative Binomial distribution is also known as Geometric distribution.

Value

A matrix containing the probability mass and the distribution function associated with each point of the support of a Negative Binomial distribution with parameters n and p.

Examples

n=3;p=0.3
plotNegBinom(n,p,type="d")
plotNegBinom(n,p,type="p",col="pink")
plotNegBinom(n,p)

Density Function, Distribution Function and/or Quantile Function Representations associated with a Normal Distribution

Description

plotNorm represents density, distribution and/or quantile functions associated with a Normal distribution with certain parameters mu and sigma.

Usage

plotNorm(mu, sigma, type = "b", col = "black")

Arguments

Value

This function is called for the side effect of drawing the plot.

Examples

mu=10; sigma=5
plotNorm(mu,sigma)
plotNorm(mu,sigma,col="red")
plotNorm(mu,sigma,type="q")
plotNorm(mu,sigma,type="dis")
plotNorm(mu,sigma,type="den")

Probability Mass and/or Distribution Function Representations associated with a Poisson Distribution

Description

plotPois represents the probability mass and/or the distribution function associated with a Poisson distribution with parameter lambda.

Usage

plotPois(lambda, type = "b", col = "grey")

Arguments

Value

A matrix containing the probability mass and the distribution function associated with each point of the support of a Poisson distribution with parameter lambda.

Examples

lambda=2
plotPois(lambda,type="d")
plotPois(lambda,type="p",col="pink")
plotPois(lambda)

Representation of a Linear Regression Model

Description

Representation of a Linear Regression Model

Usage

plotReg(x, y, main = "Linear Regression Model",
  xlab = "Explanatory variable (X)", ylab = "Response variable (Y)",
  col.points = "black", col.line = "red", pch = 19, lwd = 2,
  legend = TRUE)

Arguments

Value

This function is called for the side effect of drawing the plot.

Examples

x=rnorm(100)
error=rnorm(100)
y=1+5*x+error
plotReg(x,y)

Density Function, Distribution Function and/or Quantile Function Representations associated with a T-Student Distribution

Description

plotTS represents density, distribution and/or quantile functions associated with a T-Student distribution with df degrees of freedom.

Usage

plotTS(df, type = "b", col = "black")

Arguments

Value

This function is called for the side effect of drawing the plot.

Examples

df=10
plotTS(df)
plotTS(df,col="red")
plotTS(df,type="q")
plotTS(df,type="dis")
plotTS(df,type="den")

Density Function, Distribution Function and/or Quantile Function Representations associated with a Uniform Distribution

Description

plotUnif represents density, distribution and/or quantile functions associated with a Uniform distribution with min and max the lower and upper limits, respectively.

Usage

plotUnif(min, max, type = "b", col = "black")

Arguments

Value

This function is called for the side effect of drawing the plot.

Examples

min=0 ; max=1
plotUnif(min,max)
plotUnif(min,max,col="red")
plotUnif(min,max,type="q")
plotUnif(min,max,type="dis")
plotUnif(min,max,type="den")

Large Sample Confidence Interval for a Population Proportion

Description

proportion.CI provides a pointwise estimation and a confidence interval for a population proportion.

Usage

proportion.CI(x, n, conf.level)

Arguments

Details

Counts of successes and failures must be nonnegative and hence not greater than the corresponding numbers of trials which must be positive. All finite counts should be integers.

If the number of successes are given, then the proportion estimate is computed.

Value

A list containing the following components:

Independently on the user saving those values, the function provides a summary of the result on the console.

Examples

#Given the sample proportion estimate
proportion.CI(0.3, 100, conf.level=0.95)

#Given the number of successes
proportion.CI(30,100,conf.level=0.95)

Large Sample Test for a Population Proportion

Description

proportion.test allows to compute a hypothesis test for a population proportion.

Usage

proportion.test(x, n, p0, alternative = "two.sided", alpha = 0.05,
  plot = TRUE, lwd = 1)

Arguments

Details

Counts of successes and failures must be nonnegative and hence not greater than the corresponding numbers of trials which must be positive. All finite counts should be integers. If the number of successes is given, then the proportion estimate is computed.

Value

A list with class "lstest" and "htest" containing the following components:

Examples

x <- rbinom(1, 120, 0.6)
proportion.test(x, 120, 0.5, alternative = "greater")
proportion.test(0.6, 120, 0.5, alternative = "greater")

Data Input

Description

read.data allows to read a file and create a data frame from it. Wrapper for different data input functions available, namely data.table::fread, readxl::read_excel, haven::read_sas, haven::read_sav, haven::read_dta and readODS::read_ods. The file extensions supported are: csv, dat, data, dta, ods, RDa, RData, sas7bdat, sav, txt, xls and xlsx.

Usage

read.data(name, dec = ".", header = "auto", sheet = 1, ...)

Arguments

Value

A data.frame containing the data in the specified file or, if Rdata or Rda, an object of class "ls_str".

Examples

data <- data.frame(V1 = 1:5*0.1, V2 = 6:10)
tf <- tempfile("tp", fileext = c(".csv", ".txt", ".RData"))

write.csv(data, tf[1], row.names = FALSE)
read.data(tf[1])                             # csv
write.csv2(data, tf[1], row.names = FALSE)
read.data(tf[1], dec = ",")                  # csv2
write.table(data, tf[2], row.names = FALSE)
read.data(tf[2])                             # txt
save(data, file = tf[3])
read.data(tf[3])                             # RData

Sample Quantiles

Description

sample.quantile computes a estimation of different quantiles, given a sample x, using order statistics.

Usage

sample.quantile(x, tau)

Arguments

Details

A quantile tau determines the proportion of values in a distribution are above or below a certain limit. For instance, given tau a number between 0 and 1, the tau-quantile splits the sample into tow parts with probabilities tau and (1-tau), respectively.

One possible way to calculate the quantile tau would be to ordering the sample and taking as the quantile the smallest data in the sample (first of the ordered sample) whose cumulative relative frequency is greater than tau. If there is a point in the sample with a cumulative relative frequency equal to tau, then the sample quantile will be calculated as the mean between that point and the next one of the ordered sample.

Value

A number or a numeric vector of tau-quantile(s).

A numerical value or vector corresponding with the requested sample quantiles.

Examples

x=rnorm(20)
sample.quantile(x,tau=0.5)
sample.quantile(x,tau=c(0.25,0.5,0.75))

Sample Standard Deviation

Description

sample.sd computes the sample standard deviation of a sample x.

Usage

sample.sd(x)

Arguments

Details

Given \{x_1,\ldots,x_n\} a sample of a random variable, the sample standard deviation can be computed as S=\sqrt{\frac{1}{n}\sum_{i=1}^n (x_i-\bar{x})^2}.

Value

A single numerical value corresponding with the sample standard deviation.

Examples

x=rnorm(20)
sample.sd(x)

Sample Variance

Description

sample.var computes the sample variance of a sample x.

Usage

sample.var(x)

Arguments

Details

Given \{x_1,\ldots,x_n\} a sample of a random variable, the sample variance can be computed as S^2=\frac{1}{n}\sum_{i=1}^n (x_i-\bar{x})^2.

Value

A single numerical value corresponding with the sample variance.

Examples

x=rnorm(20)
sample.var(x)

SICRI: information system on risk-taking behaviour

Description

SICRI collects data on health-related behaviours, as long as they do not involve questions that can be considered sensitive or delicate, since with them simple self-declaration is not a way to obtain valid information. However, although the data collected will mainly refer to behaviours, data of another type may be collected for which simple self-declaration is an accommodated mode of measurement.

Usage

data(sicri2018)

Format

sicri2018 is a data frame with 7853 cases (rows) and 18 variables (columns) selected from the original and complete data base. The selection was done to have an overview of different types of random variables, as well as the topic interest for the students to discuss about.

In what follows the explanation and codification of the different variables is detailed. For every variable, we provide its name on the data base, the explanation of its content and all the possible categories, whether it makes sense, with their codes on the data base between brackets.

Source

This data has been collected from SERGAS (Galician Health Service) Database on February 2021 https://www.sergas.es/Saude-publica/SICRI-2018-Microdatos

Examples

data(sicri2018)
bmi <- sicri2018$bmi
Histogram(bmi,col="pink")

Confidence Interval for the Variance and the Standard Deviation of a Normal Population

Description

variance.CI provides a pointwise estimation and a confidence interval for the variance and the standard deviation of a normal population in both scenarios: known and unknown population mean.

Usage

variance.CI(x = NULL, s = NULL, sc = NULL, smu = NULL, mu = NULL,
  n = NULL, conf.level)

Arguments

Details

The formula interface is applicable when the user provides the sample and also when the user provides the value of sample characteristics (cuasi-standard deviation or sample standard deviation and the sample size).

Value

A list containing the following components:

Independently on the user saving those values, the function provides a summary of the result on the console.

Examples

#Given the estimation of the standard deviation with known population mean
dat=rnorm(20,mean=2,sd=1)
smu=Smu(dat,mu=2)
variance.CI(smu=smu,mu=2,n=20,conf.level=0.95)

#Given the sample with known population mean
dat=rnorm(20,mean=2,sd=1)
variance.CI(dat,mu=2,conf.level=0.95)

#Given the sample with unknown population mean
dat=rnorm(20,mean=2,sd=1)
variance.CI(dat,conf.level=0.95)

#Given the cuasi-standard deviation with unknown population mean
dat=rnorm(20,mean=2,sd=1)
variance.CI(sc=sd(dat),n=20,conf.level=0.95)

One Sample Variance Test of a Normal Population

Description

variance.test allows to compute hypothesis tests for the variance of a Normal population in both scenarios: known or unknown population mean.

Usage

variance.test(x = NULL, s = NULL, sc = NULL, smu = NULL, mu = NULL,
  n = NULL, sigma02, alternative = "two.sided", alpha = 0.05,
  plot = TRUE, lwd = 1)

Arguments

Details

The formula interface is applicable when the user provides the sample or values of the sample characteristics (cuasi-standard deviation or sample standard deviation).

Value

A list with class "lstest" and "htest" containing the following components:

Examples

x <- rnorm(50, mean = 1, sd = 2)
# unknown population mean
variance.test(x, sigma02 = 3.5)
variance.test(sc = sd(x), n = 50, sigma02 = 3.5)
# known population mean
variance.test(x, sigma02 = 3.5, mu = 1)
smu <- Smu(x, mu = 1)
variance.test(smu = smu, n = 50, sigma02 = 3.5)