Adv Quant: General Linear Regression Model in R

Converting the dataset to a dataframe for analysis and performing a regression on this dataset.

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Introduction

A goal for this post is to convert the dataset to a dataframe for analysis and performing a regression on the state.x77 dataset.

Results

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Figure 1: Scatter plot matrix of the dataframe state.x77.  The red box illustrates the relationship that is personally identified for further analysis.

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Figure 2: Scatter plot of murder rates versus illiteracy rates across the united states, with the linear regression function of illiteracy = 0.11607 * Murder + 0.31362; with a correlation of 0.729752.

Discussion

This post analyzes the dataset state.x77 under the MASS R library, was converted into a data frame (see code section), and an analysis of the data was conducted.  To identify which variable relationship would be interesting to conduct a regression on this dataset, all the relationships within the data frame were plotted in a matrix (Figure 1).  The relationship that personally seemed interesting was the relationship between illiteracy and murder.  Thus, moving forward with these variables a simple linear regression was conducted on that data.  It was determined that there is a positive correlation on this data of 0.729752, and the relationship between the data is defined by

illiteracy = 0.11607 * Murder + 0.31362                                        (1)

From this equation that describes the relationship (Figure 2) between these variables, can explain, 53.25% of the variance between these variables. Both the intercept value and the regression weight are statistically significant at the 0.01 level, meaning that there is less than a 1% chance that this relationship could be developed from pure random chance (R output between Figure 1 & 2).  In conclusion, this data is stating that states with lower illiteracy rates will have the least amount of murder rates in their state, and vice versa. 

Code

#

## Converting a dataset to a dataframe for analysis.

#

library(MASS)             # Activate the MASS library

library(nutshell)         # Activate the nutshell library to access the plot function

data()                    # Lists all data and datasets within the Mass Library

data(state)               # Data in question is located in state

head(state.x77)           # Print out the top five entries of state.x77

df= data.frame(state.x77) # Convert the state.x77 data into a dataframe

#

## Regression formulation

#

plot(df)                                           # Scatter plot matrix, of all relationships between the variables in the df

stateRegression = lm(Illiteracy~Murder, data= df)  # Selecting this relationship for further analysis

summary(stateRegression)                           # Plotting a summary of the regression data

# Plotting a scatterplot from a dataframe below

plot(df$Murder, df$Illiteracy, type=”p”, main=”Illiteracy rates vs Murder rates”, xlab=”Murder”, ylab=”Illiteracy”)           # Plotting a scatterplot from a dataframe

abline(lm(Illiteracy~Murder, data= df), col=”red”) # Plotting a red regression line

cor(df$Murder, df$Illiteracy)

References

Adv Quant: General Least Squares Model

This post compares the assumptions of General Least Square Model (GLM) modeling for regression and correlations and it covers the issues with transforming variables to make them linear.

Regression formulas are useful for summarizing the relationship between the variables in question (Huck, 2011). There are multiple types of regression all of them are tests of prediction (Huck, 2011; Schumacker, 2014).  The least squares (linear) regression is the most well-known because it uses basic algebra, a straight line, and the correlation coefficient to aid in stating the regression’s prediction strength (Huck, 2011; Schumacker, 2014).  The linear regression model is:

y = (a + bx) + e                                                                   (1)

Where y is the dependent variable, x is the independent variable, a (the intercept) and b (the regression weight, also known as the slope) are a constants that are to be defined through the regression analysis, and e is the regression prediction error (Field, 2013; Schumacker, 2014).  The sum of the squared errors should be minimized per the least squares criterion, and that is reflected in the b term in equation 1 (Schumacker, 2014).

Correlation coefficients help define the strength of the regression formula in defining the relationships between the variables, and can vary in value from -1 to +1.  The closer the correlation coefficient is to -1 or +1; it informs the researcher that the regression formula is a good predictor of the variance between the variables.  The closer the correlation coefficient is to zero, indicates that there is hardly any relationship between the variable (Field, 2013; Huck, 2011; Schumacker, 2014).  Correlations never imply causation, but they can help determine the percentage of the variances between the variables by the regression formula result when the correlation value is squared (r2) (Field, 2013).

Assumptions for the General Least Square Model (GLM) modeling for regression and correlations

The General Least Squares Model (GLM) is the line of best fit, for linear regressions modeling along with its corresponding correlations (Smith, 2015).  There are five assumptions to a linear regression model: additivity, linearity, independent errors, homoscedasticity, and normally distributed errors.  Variables should be linearly related the independent variables(s), and the combined effects of multiple independent variables should be additive. A residual is the difference between the predicted value from the observed value: (1) no two residuals should be correlated, which can be numerically tested by using the Durbin-Watson test; (2) the variance of these residuals should be constant for each independent variable; and (3) the residuals should be random and normally distributed with a mean of 0 (Field, 2013; Schumacker, 2014).

Covering the issues with transforming variables to make them linear

When viewing the data through scatter plots, if the linearity and additivity assumptions could not be met, then transformations to the variables could be made to make the relationship linear. The above is an iterative trial and error process.  Transformation must occur to every point of the data set to correct for the linearity and addititvity issues since it changes the difference between the variables due to the change of units in the variables (Field, 2013).

Table 1: Types of data transformations and their uses (adapted from Field (2013) Table 5.1).

Data Transformation Can Correct for
Log [independent variable(s)] Positive skew, positive kurtosis, unequal variances, lack of linearity
Square root [independent variable(s)] Positive skew, positive kurtosis, unequal variances, lack of linearity
Reciprocal [independent variable(s)] Positive skew, positive kurtosis, unequal variances
Reverse score [independent variable(s)]: subtracting the highest value in the variable for each data set Negative skew

Describe the R procedures for linear regression

lm( ) is a function for running linear regression, glm( ) is a function for running logistic regression (should not be confused for GLM), and loglm( ) is a function for running log-linear regression in R (Schumacker, 2014; Smith, 2015). The summary( ) function is used to output the results of the linear regression. Dependent variables are represented with a tilde “~” and independent variables are represented with a “+” (Schumacker, 2014). Thus, the R procedures for linear regression are (Marin, 2013):

> cor (x, y) # correlation coefficient

> myRegression = lm (y ~ x, data = dataSet ) # conduct a linear regression on x and y

> summary(myRegression) # produces the outputs of the lm( ) function calculations

> attributes(myRegression) # lists the attributes of the lm( ) function

> myRegression$coefficients # gives you the slope and intercept coefficients

> plot (x, y, main=“Title to graph”) # scatter plot

> abline(myRegression) # regression line

> confint(myRegression, level= 0.99) # 99% level of confidence intervals for the regression coefficients

> anova(myRegression) # anova analysis on the regression analysis

References

  • Field, A. (2013) Discovering Statistics Using IBM SPSS Statistics (4th ed.). UK: Sage Publications Ltd. VitalBook file.
  • Huck, S. W. (2011) Reading Statistics and Research (6th ed.). Pearson Learning Solutions. VitalBook file.
  • Marin, M. (2013) Linear regression in R (R tutorial 5.1). Retrieved from https://www.youtube.com/watch?v=66z_MRwtFJM
  • Schumacker, R. E. (2014) Learning statistics using R. California, SAGE Publications, Inc, VitalBook file.
  • Smith, M. (2015). Statistical analysis handbook. Retrieved from http://www.statsref.com/HTML/index.html?introduction.html

Adv Quant: Birth Rate Dataset in R

Built in the R library is the Births dataset with 400,000 records and 13 variables. The following is an analysis of this dataset.

Introduction

Built in the R library is the Births dataset with 400,000 records and 13 variables.  The following is an analysis of this dataset.

Results

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Figure 1. The first five data point entries in the births2006.smpl data set.

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Figure 2. The frequency of births in 2006 per day of the week.

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Figure 3. Histogram of 2006 births frequencies graphed by day of the week and separated by method of delivery.

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Figure 4. A trellis histogram plot of 2006 birth weight per birth number.

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Figure 5. A trellis histogram plot of 2006 birth weight per birth delivery method.

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Figure 6. A boxplot of 2006 birth weight per Apgar score.

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Figure 7. A boxplot of 2006 birth weight per day of week.

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Figure 8. A histogram of 2006 average birth weight per multiple births separated by gender.

Discussion

Given the open-sourced nature of the R software, many libraries are being built and shared with the greater community, and the Comprehensive R Archive Network (CRAN), has a ton of these programs as part of R Packages (Schumacker, 2014).  Thus, as part of the nutshell library, there exists a data set of 2006 births called “births2006.smpl”.  To view the first few entries the head() command can be used (R, n.d.g.).  The printout from the head() command (Figure 1) shows all 13 variables of the dataset along with the first five entries in the births2006.smpl dataset.

The number of birth seems to be approximately uniform (but not precisely) during the work week, assuming Sunday is 1 and Saturday is 7.  However, Tuesday-Thursday has the highest births in the week with the weekends having the least amount of births in the week.

Breaking down the method of deliveries in 2006 per day of the week, it can be seen that Vaginal birth in all seven days of the week outnumbers C-section deliveries in 2006 (Figure 3).  Also on Tuesday-Thursday there are more vaginal births compared to those during the weekend, and in C-section deliveries, there are most deliveries occur between Tuesday-Friday, and the least amount occurs during the weekends.

Breaking down the number of births frequencies per birth weight (Figure 4), it can be seen that the normal distribution of birth weight in grams shifts to the left as the number of multiple births increases.  This seems to suggest that babies born as a set of twins, triplets, etc. have lower birth rates on average and per distribution.  Birth weight is almost normally distributed for the single child birth but begins to lose normality as the number of births increases.

Further analysis of birth weights in 2006, per delivery method, shows that for whether or not the delivery method is known or not and its type of delivery method doesn’t play too much of a huge role in the determination of the child’s birth weight (Figure 5).  Statistical tests and effect size analysis could be conducted to verify and enhance the discussion and this assertion that is made through the graphical representation in Figure 5.

Apgar test is tested on the child after one and five minutes of birth looking at the skin color, heart rate, reflexes, muscle tone, and respiration rate of the child, where 10 is the highest but rarely obtain score (Hirsch, 2014).  Thus, observing the Apgar score variable (1-10) on birth weight in grams those with higher Apgar scores had on average higher median birth weights.  Typically, as Apgar score increases the tighter the distribution becomes, and the more outliers begin to appear (disregarding the results from Apgar score of 1).  These results from the boxplots tend to confirm Hirsch (2014) assertion that higher Apgar scores are harder to obtain.

Looking at the boxplot analysis of birth weight per day of the week (Figure 7) shows that the median, Q1, Q3, max, and min are normally distributed and unchanging per day of the week.  Outliers, the heavier babies, tend to occur without respect of the day of the week, and also appears to have little to no effect on the distribution of birth weight per day of the week.

Finally, looking at a mean birth weight per gender and per multiple births, shows a similar distribution of males and females (Figure 8). The main noticeable difference is the male Quintuplet or higher number of births on average weigh more than the corresponding female Quintuplet or higher number of births.  This chart also confirms the conclusions made (from Figure 4) where as the number of births increases the average weight of the children decrease.

In conclusion, the day of the week doesn’t predict birth weights, but probably birth frequency. In general, babies are heavier if they are single births and if they achieve Apgar score of 10.  Birth weights are not predictable through delivery method.  All of these conclusions are made on the visual representation of the dataset births2006.smpl.  What would increase the validity of these statements would be to conduct statistical significance tests and the effect size, to add further weight to what could be derived from through these images.

Code

#
## Use R to analyze the Birth dataset. 
## The Birth dataset is in the Nutshell library. 
##  • SEX and APGAR5 (SEX and Apgar score) 
##  • DPLURAL (single or multiple birth) 
##  • WTGAIN (weight gain of mother) 
##  • ESTGEST (estimated gestation in weeks) 
##  • DOB_MM, DOB_WK (month and day of week of birth) 
##  • BWT (birth weight) 
##  • DMETH_REC (method of delivery)
#
install.packages(“nutshell”)
library(nutshell)
data(births2006.smpl)

# First, list the data for the first 5 births. 
head(births2006.smpl)

# Next, show a bar chart of the frequencies of births according to the day of the week of the birth.
births.dayofweek = table(births2006.smpl$DOB_WK) #Goal of this variable is to speed up the calculations
barplot(births.dayofweek, ylab=”frequency”, xlab=”Day of week”, col = “darkred”, main= “Number of births in 2006 per day of the week”)

# Obtain frequencies for two-way classifications of birth according to the day of the week and the method of delivery.
births.methodsVdaysofweek = table(births2006.smpl$DOB_WK,births2006.smpl$DMETH_REC) 
head(births.methodsVdaysofweek,7)
barplot(births.methodsVdaysofweek[,-2], col=heat.colors(length(rownames(births.methodsVdaysofweek))), width=2, beside=TRUE, main = “bar plot of births per method per day of the week”)
legend (“topleft”, fill=heat.colors(length(rownames(births.methodsVdaysofweek))),legend=rownames(births.methodsVdaysofweek))

# Use lattice (trellis) graphs (R package lattice) to condition density histograms on the values of a third variable. 
library(lattice)

# The variable for multiple births and the method of delivery are conditioning variables. 
# Separate the histogram of birth weight according to these variable.
histogram(~DBWT|DPLURAL,data=births2006.smpl,layout=c(1,5),col=”black”, xlab = “birth weight”, main = “trellis plot of birth weight vs birth number”)

histogram(~DBWT|DMETH_REC,data=births2006.smpl,layout=c(1,3),col=”black”, xlab = “birth weight”, main = “trellis plot of birth weight vs birth method”)

# Do a box plot of birth weight against Apgar score and box plots of birth weight by day of week of delivery. 
boxplot(DBWT~APGAR5,data=births2006.smpl,ylab=”birth weight”,xlab=”AGPAR5″, main=”Boxplot of birthweight per Apgar score”)

boxplot(DBWT~DOB_WK,data=births2006.smpl,ylab=”birth weight”,xlab=”Day of Week”, main=”Boxplot of birthweight per day of week”)

# Calculate the average birth weight as a function of multiple births for males and females separately. 
# Use the “tapply” function, and for missing values use the “option nz.rm=TRUE.” 
listed = list(births2006.smpl$DPLURAL,births2006.smpl$SEX)
tapplication=tapply(births2006.smpl$DBWT,listed,mean,na.rm=TRUE)
barplot(tapplication,ylab=”birth weight”, beside=TRUE, legend=TRUE,xlab=”gender”, main = “bar plot of average birthweight per multiple births by gender”)

References

  • CRAN (n.d.). Using lattice’s historgram (). Retrieved from https://cran.r-project.org/web/packages/tigerstats/vignettes/histogram.html
  • Hirsch, L. (2014). About the Apgar score. Retrieved from http://kidshealth.org/en/parents/apgar.html#
  • R (n.d.a.). Add legends to plots. Retrieved from https://stat.ethz.ch/R-manual/R-devel/library/graphics/html/legend.html
  • R (n.d.b.). Apply a function over a ragged array. Retrieved from https://stat.ethz.ch/R-manual/R-devel/library/base/html/tapply.html
  • R (n.d.c.). Bar plots. Retrieved from https://stat.ethz.ch/R-manual/R-devel/library/graphics/html/barplot.html
  • R (n.d.d.). Cross tabulation and table creation. Retrieved from https://stat.ethz.ch/R-manual/R-devel/library/base/html/table.html
  • R (n.d.e.). List-Generic and dotted pairs. Retrieved from https://stat.ethz.ch/R-manual/R-devel/library/base/html/list.html
  • R (n.d.f.). Produce box-and-wisker plot(s) of a given (grouped) values.  Retrieved from https://stat.ethz.ch/R-manual/R-devel/library/graphics/html/boxplot.html
  • R (n.d.g.). Return the first or last part of an object. Retrieved from https://stat.ethz.ch/R-manual/R-devel/library/utils/html/head.html
  • Schumacker, R. E. (2014) Learning statistics using R. California, SAGE Publications, Inc, VitalBook file.

Adv Quant: Statistical Significance and Machine Learning

Data mining and analytics are used to test hypotheses and detect trends from very large datasets. In statistics, the significance is determined to some extent by the sample size. How can supervised learning be used in such large data sets to overcome the problem where everything is significant with statistical analysis?

Statistical significance on large samples sizes can be affected by small differences and can show up as significant, while in smaller samples large differences may be deemed statistically insignificant (Field, 2014).  Statistically significant results allow the researcher to reject a null hypothesis but do not test the importance of the observations made (Huck, 2011). Statistical analysis is highly deductive (Creswell, 2014), and supervised learning is highly inductive (Connolly & Begg, 2014).  Also, statistical analysis tries to identify trends in a given sample size by assuming normality, linearity or constant variance; whereas in machine learning it aims to find a pattern in a large sample of data and it is expected that these statistical analysis assumptions are not met and therefore require a higher random sampling set (Ahlemeyer-Stubbe, & Coleman, 2014).

Machine learning tries to emulate the way humans learn. When humans learn, they create a model based off of observations to help describe key features of a situation and help them predict an outcome, and thus machine learning does predictive modeling of large data sets in a similar fashion (Connolly & Begg, 2014).  The biggest selling point of supervised machine learning is that the machine can build models that identify key patterns in the data when humans can no longer compute the volume, velocity, and variety of the data (Ahlemeyer-Stubbe, & Coleman, 2014). There are many applications that use machine learning: marketing, investments, fraud detection, manufacturing, telecommunication, etc. (Fayyad, Piatetsky-Shapiro, & Smyth, 1996). Figure 1 illustrates how supervised learning can classify data or predict their values through a two-phase process.  The two-phase process consists of (1) training where the model is built by ingesting huge amounts of historical data; and (2) testing where the new model is tested on new current data that helps establish its accuracy, reliability, and validity (Ahlemeyer-Stubbe & Coleman, 2014; Connolly & Begg, 2014). The model that is created by machines through this learning is quickly adaptable to new data (Minelli, Chambers, & Dhiraj, 2013).  These models themselves are a set of rules or formulas, and that depends on which analytical algorithm is used (Ahlemeyer-Stubbe & Coleman, 2014).  Given that the supervised machine learning is trained with known responses (or outputs) to make its future predictions, it is vital to have a clear purpose defined before running the algorithm.  The model is only as good as the data that goes in it.

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Figure 1:  Simplified process diagram on supervised machine learning.

Thus, for classification the machine is learning a function to map data into one or many different defining characteristics, and it could consist of decision trees and neural network induction techniques (Connolly & Begg, 2014; Fayyad et al., 1996).  Fayyad et al. (1996) mentioned that it is impossible to classify data cleanly into one camp versus another. For value prediction, regression is used to map a function to the data that when followed gives an estimate on where the next value would be (Connolly & Begg, 2014; Fayyad et al. 1996).  However, in these regression formulas, it is good to remember that correlation between the data/variables does not imply causation.

Random sampling is core to statistics and the concept of statistical inference (Smith, 2015; Field, 2011), but it also serves a purpose in supervised learning (Ahlemeyer-Stubbe & Coleman, 2014).  Random sampling of data, is selecting a proportion of the data from a population, where each data point has an equal opportunity of being selected (Smith, 2015; Huck, 2013). The larger the sample, on average tends to represent the population fairly well (Field, 2014; Huck, 2013). Given nature big data, high volume, velocity, and variety, it is assumed that there is plenty of data to draw upon and run a supervised machine learning algorithm.  However, too much data that is fed into the machine learning algorithm can increase the process and analysis time.  Also, the bigger the random sampling size used for the learning, the more time it would take to process and analyze the data.

There are also unsupervised learning algorithms, where it also needs training and testing, but unlike supervised learning, it doesn’t need to validate its model on some predetermined output value (Ahlemeyer-Stubbe & Coleman, 2014, Conolly & Begg, 2014).   Therefore, unsupervised learning tries to find the natural relationships in the input data (Ahlemeyer-Stubbe & Coleman, 2014).  Cluster analysis is an example of unsupervised learning, where the model seeks to find a finite set of the cluster that can help describe the data into subsets of similarities (Ahlemeyer-Stubbe & Coleman, 2014, Fayyad et al., 1996). Finally, in supervised learning the results could be checked through estimation error; however it is not so easy with unsupervised learning because of a lack of a target but requires retesting to see if the patterns are similar or repeatable (Ahlemeyer-Stubbe & Coleman, 2014).

References

  • Ahlemeyer-Stubbe, A., & Coleman, S. (2014). A Practical Guide to Data Mining for Business and Industry, 1st Edition. [VitalSource Bookshelf Online].
  • Connolly, T., Begg, C. (2014). Database Systems: A Practical Approach to Design, Implementation, and Management, 6th Edition. [VitalSource Bookshelf Online].
  • Creswell, J. W. (2014) Research design: Qualitative, quantitative and mixed method approaches (4th ed.). California, SAGE Publications, Inc. VitalBook file.
  • Fayyad, U., Piatetsky-Shapiro, G., & Smyth, P. (1996). From data mining to knowledge discovery in databases. Advances in Knowledge Discovery and Data Mining, 17(3), 37–54.
  • Field, A. (2011) Discovering Statistics Using IBM SPSS Statistics (4th ed.). UK: Sage Publications Ltd. VitalBook file.
  • Huck, S. W. (2013) Reading Statistics and Research (6th ed.). Pearson Learning Solutions. VitalBook file.
  • Minelli, M., Chambers, M., Dhiraj, A. (2013). Big Data, Big Analytics: Emerging Business Intelligence and Analytic Trends for Today’s Businesses, 1st Edition. [VitalSource Bookshelf Online].
  • Smith, M. (2015). Statistical analysis handbook. Retrieved from http://www.statsref.com/HTML/index.html?introduction.html