Summary

Specialisation	Data Analysis and Interpretation
Course	Regression Modeling in Practice
Education Institution	Wesleyan University
Publisher	Coursera
Assignment	Test a Basic Linear Regression Model

Introduction

Besides the historical fascination on Mars due to its proximity and some similarities with Earth, the collection of facts allow scientists to put together a big jigsaw puzzle.

An announcement made by NASA just recently about evidence of flowing liquid water on the surface of Mars just adds to all that is known and the curiosity, or even need, to find out much more.

The Data Set

With all the talks about Mars, including both Science and Fiction (a Ridley Scott’s movie called The Martian, based on the book with the same name, by Andy Weir was released on early Oct/15), the data set chosen for this assignment is the Mars Craters.

The Mars Craters Study, presents a global database that includes 378,540 Mars craters, with diameter of 1 km or larger, that were created between 4.2 and 3.8 billion years ago during a period of heavy bombardment (i.e. impacts of asteroids, proto-planets, and comets).

The data set was made available by Wesleyan University/Coursera as part of the Data Analysis and Interpretation Specialisation, from the Ph.D. Thesis Planetary Surface Properties, Cratering Physics, and the Volcanic History of Mars from a New Global Martian Crater Database (2011) by Robbins, S.J., University of Colorado at Boulder.

Topic of Interest

The data set provides a catalogue of craters on Mars. The initial thoughts are about checking for patterns that could identify specific major events that might have happened and that would have significant impact on Mars’ geology, climate and life as a planetary body.

Codebook

As the initial data set has only nine variables, they could all the relevant to formulate hypothesis and help in leading to a conclusion, so all the variables will be kept for this assignment.

Variables

CRATER_ID: crater ID for internal sue, based upon the region of the planet (\({1 \over 16}\)), the “pass” under which the crate was identified, ad the order in which it was identified
LATITUDE_CIRCLE_IMAGE: latitude from the derived centre of a non-linear least-squares circle fit to the vertices selected to manually identify the crater rim (units are decimal degrees North)
LONGITUDE_CIRCLE_IMAGE: longitude from the derived centre of a non-linear least-squares circle fit to the vertices selected to manually identify the crater rim (units are decimal degrees East)
DIAM_CIRCLE_IMAGE: diameter from a non-linear least squares circle fit to the vertices selected to manually identify the crater rim (units are km)
DEPTH_RIMFLOOR_TOPOG: average elevation of each of the manually determined N points along (or inside) the crater rim (units are km)
- Depth Rim: Points are selected as relative topographic highs under the assumption they are the least eroded so most original points along the rim
- Depth Floor: Points were chosen as the lowest elevation that did not include visible embedded craters
MORPHOLOGY_EJECTA_1: ejecta morphology classified.
- If there are multiple values, separated by a “/”, then the order is the inner-most ejecta through the outer-most, or the top-most through the bottom-most
MORPHOLOGY_EJECTA_2: the morphology of the layer(s) itself/themselves. This classification system is unique to this work.
MORPHOLOGY_EJECTA_3: overall texture and/or shape of some of the layer(s)/ejecta that are generally unique and deserve separate morphological classification.
NUMBER_LAYERS: the maximum number of cohesive layers in any azimuthal direction that could be reliably identified

Test a Basic Linear Regression Model

Is the Diameter of the creaters associated with their Depth?

Variables

In the original full database:

Diameter (DIAM_CIRCLE_IMAGE) is a quantitative variable that ranges from 1.00 to 1,164.22. It will be used as the dependent variable in this test.
Depth (DEPTH_RIMFLOOR_TOPOG) is a quantitative variable that ranges from -0.42 to 4.95. It will be used as the explanatory variable in this test.

Sample

A sample of about ten percent of the original data set, with 38,435 out of 378,540 cases, will be used for the regression.

Outliers

There are two cases which could be considered outliers based on the numbers despite the correctness of the data. These two cases no not alter considerably the regression model when compared to the one produced without removing the outliers.

Regression

In both scenarios with and without the outliers the p-value is significatly lower than the alpha value of five percent; \(\alpha = 0.05\) and \(p{-}value < 0.0001\).

As per the regression model, for every extra km in Depth there would be an increase of 22.76 km in Diameter, considering the Sample Data.

This indicates a significant and positive association between Diameter and Depth.

\(Diameter = 3.61806635 + 22.76468952 * Depth\) (Sample Data)

\(Diameter = 3.57347150 + 22.83346222 * Depth\) (Outliers removed)

Using SAS

SAS Output

Results: W02-Test a Basic Linear Regression Model-Local.sas

The SURVEYSELECT Procedure

Sample Selection Method

Selection Method	Simple Random Sampling

Sample Selection Summary

Input Data Set	WORK
Random Number Seed	196587
Sampling Rate	0.1
Sample Size	38435
Selection Probability	0.100002
Sampling Weight	9.999818
Output Data Set	EXTRACT

Mars' Craters - Summary - Sample Data

The SUMMARY Procedure

Summary statistics

Variable	Label	N	Mean	Std Dev	Minimum	Maximum
DEPTH_RIMFLOOR_TOPOG Depth DIAM_CIRCLE_IMAGE	Depth Diameter	38435 38435 38435	0.0756717 -3.46074E-16 3.6180663	0.2217400 0.2217400 10.3805219	-0.0100000 -0.0856717 1.0000000	3.6000000 3.5243283 1096.65

The SGPLOT Procedure

The SGPlot Procedure

Mars' Craters - GLM - Sample Data

The GLM Procedure

Data

Number of Observations

Number of Observations Read	38435
Number of Observations Used	38435

Mars' Craters - GLM - Sample Data

The GLM Procedure

Dependent Variable: DIAM_CIRCLE_IMAGE Diameter

Analysis of Variance

DIAM_CIRCLE_IMAGE

Overall ANOVA

Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	1	979325.302	979325.302	11902.8	<.0001
Error	38433	3162139.359	82.277
Corrected Total	38434	4141464.661

Fit Statistics

R-Square	Coeff Var	Root MSE	DIAM_CIRCLE_IMAGE Mean
0.236468	250.7043	9.070649	3.618066

Type I Model ANOVA

Source	DF	Type I SS	Mean Square	F Value	Pr > F
Depth	1	979325.3017	979325.3017	11902.8	<.0001

Type III Model ANOVA

Source	DF	Type III SS	Mean Square	F Value	Pr > F
Depth	1	979325.3017	979325.3017	11902.8	<.0001

Solution

Parameter	Estimate	Standard Error	t Value	Pr > \|t\|
Intercept	3.61806635	0.04626738	78.20	<.0001
Depth	22.76468952	0.20865875	109.10	<.0001

Mars' Craters - Summary - Outliers removed

The SUMMARY Procedure

Summary statistics

Variable	Label	N	Mean	Std Dev	Minimum	Maximum
DEPTH_RIMFLOOR_TOPOG Depth DIAM_CIRCLE_IMAGE	Depth Diameter	38433 38433 38433	0.0756756 -6.21191E-17 3.5734715	0.2217451 0.2217451 8.1634387	-0.0100000 -0.0856756 1.0000000	3.6000000 3.5243244 326.7700000

The SGPLOT Procedure

The SGPlot Procedure

Mars' Craters - GLM - Outliers removed

The GLM Procedure

Data

Number of Observations

Number of Observations Read	38433
Number of Observations Used	38433

Mars' Craters - GLM - Outliers removed

The GLM Procedure

Dependent Variable: DIAM_CIRCLE_IMAGE Diameter

Analysis of Variance

DIAM_CIRCLE_IMAGE

Overall ANOVA

Source	DF	Sum of Squares	Mean Square	F Value	Pr > F
Model	1	985245.401	985245.401	24026.4	<.0001
Error	38431	1575929.644	41.007
Corrected Total	38432	2561175.045

Fit Statistics

R-Square	Coeff Var	Root MSE	DIAM_CIRCLE_IMAGE Mean
0.384685	179.1997	6.403650	3.573471

Type I Model ANOVA

Source	DF	Type I SS	Mean Square	F Value	Pr > F
Depth	1	985245.4011	985245.4011	24026.4	<.0001

Type III Model ANOVA

Source	DF	Type III SS	Mean Square	F Value	Pr > F
Depth	1	985245.4011	985245.4011	24026.4	<.0001

Solution

Parameter	Estimate	Standard Error	t Value	Pr > \|t\|
Intercept	3.57347150	0.03266446	109.40	<.0001
Depth	22.83346222	0.14730827	155.00	<.0001

SAS Code

/* Using SAS Educational Virtual Machine running locally */
/* For CSV Files uploaded from MacOS */
FILENAME CSV     "/folders/myfolders/marscrater_pds.csv" 
         TERMSTR = CRLF;

PROC IMPORT DATAFILE = CSV
            OUT      = WORK
            DBMS     = CSV
            REPLACE;
RUN;

/* Unassign the file reference. */
FILENAME CSV;

DATA WORK;
  SET   WORK;

  LABEL DEPTH_RIMFLOOR_TOPOG = "Depth"     /* Explanatory - Quantitative */
        DIAM_CIRCLE_IMAGE    = "Diameter"; /* Response    - Quantitative */
RUN;

/* Select a sample of 10% of the population */
PROC SURVEYSELECT DATA     = WORK 
                  OUT      = EXTRACT 
                  METHOD   = SRS 
                  SAMPRATE = 0.1 
                  SEED     = 196587;
  ID DIAM_CIRCLE_IMAGE 
     DEPTH_RIMFLOOR_TOPOG;
RUN;

DATA SAMPLE ;
  SET   EXTRACT;
  Depth = DEPTH_RIMFLOOR_TOPOG - 0.0756716534408713;
RUN;

PROC SUMMARY DATA = SAMPLE PRINT ;
  VAR   DEPTH_RIMFLOOR_TOPOG
        Depth
        DIAM_CIRCLE_IMAGE;
  TITLE "Mars' Craters - Summary - Sample Data";
RUN;

PROC SGPLOT DATA = SAMPLE ;
  REG   X = Depth
        Y = DIAM_CIRCLE_IMAGE
  /     FILLEDOUTLINEDMARKERS
        MARKERATTRS        = (COLOR = GREY SIZE = 5 SYMBOL = CIRCLEFILLED)
        MARKERFILLATTRS    = (COLOR = GREY)
        MARKEROUTLINEATTRS = (COLOR = GREY THICKNESS = 1)
        LINEATTRS          = (COLOR = RED THICKNESS = 2);
  TITLE "Mars' Craters - Diameter by Depth - Sample Data";
  YAXIS LABEL = "Diameter";
  XAXIS LABEL = "Depth";
RUN;

PROC GLM DATA = SAMPLE ;
  MODEL DIAM_CIRCLE_IMAGE = Depth;
  TITLE "Mars' Craters - GLM - Sample Data";
RUN;

DATA NO_OUTLIERS ;
  SET   SAMPLE;

  WHERE DIAM_CIRCLE_IMAGE    < 500 AND
        DEPTH_RIMFLOOR_TOPOG <   4 ;

  Depth = DEPTH_RIMFLOOR_TOPOG - 0.075675591288738414;
RUN;

PROC SUMMARY DATA = NO_OUTLIERS PRINT ;
  VAR   DEPTH_RIMFLOOR_TOPOG 
        Depth 
        DIAM_CIRCLE_IMAGE;
  TITLE "Mars' Craters - Summary - Outliers removed";
RUN;

PROC SGPLOT DATA = NO_OUTLIERS;
  REG   X = Depth  
        Y = DIAM_CIRCLE_IMAGE
  /     FILLEDOUTLINEDMARKERS
        MARKERATTRS        = (COLOR = GREY SIZE = 5 SYMBOL = CIRCLEFILLED)
        MARKERFILLATTRS    = (COLOR = GREY)
        MARKEROUTLINEATTRS = (COLOR = GREY THICKNESS = 1)
        LINEATTRS          = (COLOR = RED THICKNESS = 2);
  TITLE "Mars' Craters - Diameter by Depth - Outliers removed";
  YAXIS LABEL = "Diameter";
  XAXIS LABEL = "Depth";
RUN;

PROC GLM DATA = NO_OUTLIERS;
  MODEL DIAM_CIRCLE_IMAGE = Depth;
  TITLE "Mars' Craters - GLM - Outliers removed";
RUN;
TITLE;

Mars Craters - Looking for major events

Can crater’s records help to understand Mars’ history?

Angelo Klin

26 November 2015

Summary

Introduction

The Data Set

Topic of Interest

Codebook

Variables

Test a Basic Linear Regression Model

Variables

Sample

Outliers

Regression

Using SAS

SAS Output

Results: W02-Test a Basic Linear Regression Model-Local.sas

The SURVEYSELECT Procedure

Sample Selection Method

Sample Selection Summary

The SUMMARY Procedure

Summary statistics

The SGPLOT Procedure

The SGPlot Procedure

The GLM Procedure

Data

Number of Observations

Analysis of Variance

DIAM_CIRCLE_IMAGE

Overall ANOVA

Fit Statistics

Type I Model ANOVA

Type III Model ANOVA

Solution

The SUMMARY Procedure

Summary statistics

The SGPLOT Procedure

The SGPlot Procedure

The GLM Procedure

Data

Number of Observations

Analysis of Variance

DIAM_CIRCLE_IMAGE

Overall ANOVA

Fit Statistics

Type I Model ANOVA

Type III Model ANOVA

Solution

SAS Code