Sample Selection Method
| Selection Method | Simple Random Sampling |
|---|
Mars Craters - Looking for major events
Can crater’s records help to understand Mars’ history?
Angelo Klin
14 December 2015
Summary
| Specialisation | Data Analysis and Interpretation |
| Course | Regression Modeling in Practice |
| Education Institution | Wesleyan University |
| Publisher | Coursera |
| Assignment | Test a Logistic 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.
Summary
When testing for the association between Hemisphere with Diameter and Depth using Logistic Regression the p-values are above the alpha value of 0.05, which indicates that there is no strong association between the variables when classified as binary variables.
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 Logistic Regression Model
- Is the Hemisphere of the craters associated with their Depth and Diameter?
Variables for the analysis
- North_Hemisphere is a categorical variable that has value 0 for the South hemisphere and value 1 for the North hemisphere. This is the binary categorical response variable.
- Big_Diameter is a categorical variable that splits
DIAM_CIRCLE_IMAGEby its median, with value of 0 for diameters less than 1.53 and value of 1 otherwise. - Big_Depth is a categorical variable with value of 0 for depth less than or equal to zero and value of 1 otherwise.
Sample
A sample of about one percent of the original data set, with 3,844 out of 378,540 cases, will be used for the regression.
Regression
Diameter
Although the point estimate for the Odds Ratio is 1.027, with a confidence interval of (0.902, 1.169) the p-value of 0.6969 is above the alpha value of 0.05.
As per the logistic regression model, with binary categorical variables, there is not much association between Hemisphere and Diameter.
Diameter and Depth
Although the point estimate for the Odds Ratio of diameter is 0.983, with a confidence interval of (0.847, 1.140) the p-value of 0.8205 is above the alpha value of 0.05. The same for the point estimate for the Odds Ratio of depth is 1.118, with a confidence interval of (0.927, 1.347) the p-value of 0.2426 is above the alpha value of 0.05.
As per the logistic regression model, with binary categorical variables, there is not much association between Hemisphere by Diameter and Depth.
Charts
The charts on both analysis show a ROC curve that is almost on top of the 50% threshold with an Area Under the Curve of 0.5033 for Diameter and 0.5097 for Diameter and Depth.
Using SAS
SAS Output
Results: W04-Test a Logistic Regression Model-Local.sas
The SURVEYSELECT Procedure
The SURVEYSELECT Procedure
Sample Selection Summary
| Input Data Set | WORK |
|---|---|
| Random Number Seed | 196587 |
| Sampling Rate | 0.01 |
| Sample Size | 3844 |
| Selection Probability | 0.010001 |
| Sampling Weight | 99.98517 |
| Output Data Set | EXTRACT |
The SUMMARY Procedure
The SUMMARY Procedure
Summary statistics
| Variable | Minimum | Mean | Median | Maximum |
|---|---|---|---|---|
|
DIAM_CIRCLE_IMAGE
DEPTH_RIMFLOOR_TOPOG
|
1.0000000
-0.0300000
|
3.4845239
0.0777940
|
1.5300000
0
|
297.9200000
2.4100000
|
The SUMMARY Procedure
The SUMMARY Procedure
Summary statistics
| North Hemisphere | N Obs |
|---|---|
| 0 | 2343 |
| 1 | 1501 |
The SUMMARY Procedure
The SUMMARY Procedure
Summary statistics
| Big Diameter | N Obs |
|---|---|
| 0 | 1913 |
| 1 | 1931 |
The SUMMARY Procedure
The SUMMARY Procedure
Summary statistics
| Big Depth | N Obs |
|---|---|
| 0 | 3074 |
| 1 | 770 |
The LOGISTIC Procedure
The LOGISTIC Procedure
Model Information
| Model Information | ||
|---|---|---|
| Data Set | WORK.SAMPLE | |
| Response Variable | North_Hemisphere | North Hemisphere |
| Number of Response Levels | 2 | |
| Model | binary logit | |
| Optimization Technique | Fisher's scoring | |
Observations Summary
| Number of Observations Read | 3844 |
|---|---|
| Number of Observations Used | 3844 |
Response Profile
| Response Profile | ||
|---|---|---|
|
Ordered Value |
North_Hemisphere |
Total Frequency |
| 1 | 0 | 2343 |
| 2 | 1 | 1501 |
Probability modeled is North_Hemisphere=0.
Convergence Status
| Model Convergence Status |
|---|
| Convergence criterion (GCONV=1E-8) satisfied. |
Fit Statistics
| Model Fit Statistics | ||
|---|---|---|
| Criterion | Intercept Only | Intercept and Covariates |
| AIC | 5144.978 | 5146.820 |
| SC | 5151.232 | 5159.328 |
| -2 Log L | 5142.978 | 5142.820 |
Global Tests
| Testing Global Null Hypothesis: BETA=0 | |||
|---|---|---|---|
| Test | Chi-Square | DF | Pr > ChiSq |
| Likelihood Ratio | 0.1582 | 1 | 0.6909 |
| Score | 0.1582 | 1 | 0.6909 |
| Wald | 0.1581 | 1 | 0.6909 |
Parameter Estimates
| Analysis of Maximum Likelihood Estimates | |||||
|---|---|---|---|---|---|
| Parameter | DF | Estimate |
Standard Error |
Wald Chi-Square |
Pr > ChiSq |
| Intercept | 1 | 0.4321 | 0.0468 | 85.2492 | <.0001 |
| Big_Diameter | 1 | 0.0263 | 0.0661 | 0.1581 | 0.6909 |
Odds Ratios
| Odds Ratio Estimates | |||
|---|---|---|---|
| Effect | Point Estimate |
95% Wald Confidence Limits |
|
| Big_Diameter | 1.027 | 0.902 | 1.169 |
Association Statistics
| Association of Predicted Probabilities and Observed Responses | |||
|---|---|---|---|
| Percent Concordant | 25.3 | Somers' D | 0.007 |
| Percent Discordant | 24.7 | Gamma | 0.013 |
| Percent Tied | 50.0 | Tau-a | 0.003 |
| Pairs | 3516843 | c | 0.503 |
ROC Curve
The LOGISTIC Procedure
The LOGISTIC Procedure
Model Information
| Model Information | ||
|---|---|---|
| Data Set | WORK.SAMPLE | |
| Response Variable | North_Hemisphere | North Hemisphere |
| Number of Response Levels | 2 | |
| Model | binary logit | |
| Optimization Technique | Fisher's scoring | |
Observations Summary
| Number of Observations Read | 3844 |
|---|---|
| Number of Observations Used | 3844 |
Response Profile
| Response Profile | ||
|---|---|---|
|
Ordered Value |
North_Hemisphere |
Total Frequency |
| 1 | 0 | 2343 |
| 2 | 1 | 1501 |
Probability modeled is North_Hemisphere=0.
Convergence Status
| Model Convergence Status |
|---|
| Convergence criterion (GCONV=1E-8) satisfied. |
Fit Statistics
| Model Fit Statistics | ||
|---|---|---|
| Criterion | Intercept Only | Intercept and Covariates |
| AIC | 5144.978 | 5147.451 |
| SC | 5151.232 | 5166.214 |
| -2 Log L | 5142.978 | 5141.451 |
Global Tests
| Testing Global Null Hypothesis: BETA=0 | |||
|---|---|---|---|
| Test | Chi-Square | DF | Pr > ChiSq |
| Likelihood Ratio | 1.5265 | 2 | 0.4662 |
| Score | 1.5199 | 2 | 0.4677 |
| Wald | 1.5192 | 2 | 0.4679 |
Parameter Estimates
| Analysis of Maximum Likelihood Estimates | |||||
|---|---|---|---|---|---|
| Parameter | DF | Estimate |
Standard Error |
Wald Chi-Square |
Pr > ChiSq |
| Intercept | 1 | 0.4318 | 0.0468 | 85.1401 | <.0001 |
| Big_Diameter | 1 | -0.0172 | 0.0757 | 0.0515 | 0.8205 |
| Big_Depth | 1 | 0.1114 | 0.0953 | 1.3652 | 0.2426 |
Odds Ratios
| Odds Ratio Estimates | |||
|---|---|---|---|
| Effect | Point Estimate |
95% Wald Confidence Limits |
|
| Big_Diameter | 0.983 | 0.847 | 1.140 |
| Big_Depth | 1.118 | 0.927 | 1.347 |
Association Statistics
| Association of Predicted Probabilities and Observed Responses | |||
|---|---|---|---|
| Percent Concordant | 32.0 | Somers' D | 0.019 |
| Percent Discordant | 30.1 | Gamma | 0.031 |
| Percent Tied | 37.8 | Tau-a | 0.009 |
| Pairs | 3516843 | c | 0.510 |
ROC Curve
SAS Code
1 /* Using SAS Educational Virtual Machine running locally */
2 /* For CSV Files uploaded from MacOS */
3 FILENAME CSV "/folders/myfolders/marscrater_pds.csv"
4 TERMSTR = CRLF;
5
6 PROC IMPORT DATAFILE = CSV
7 OUT = WORK
8 DBMS = CSV
9 REPLACE;
10 RUN;
11
12 /* Unassign the file reference. */
13 FILENAME CSV;
14
15 /* Select a sample of 10% of the population */
16 PROC SURVEYSELECT DATA = WORK
17 OUT = EXTRACT
18 METHOD = SRS
19 SAMPRATE = 0.01
20 SEED = 196587;
21 ID LATITUDE_CIRCLE_IMAGE
22 DIAM_CIRCLE_IMAGE
23 DEPTH_RIMFLOOR_TOPOG;
24 RUN;
25
26 PROC SUMMARY DATA = EXTRACT MIN MEAN MEDIAN MAX PRINT ;
27 VAR DIAM_CIRCLE_IMAGE
28 DEPTH_RIMFLOOR_TOPOG;
29 TITLE "Mars' Craters - Summary - Sample Data";
30 RUN;
31
32 DATA SAMPLE;
33 SET EXTRACT;
34
35 IF LATITUDE_CIRCLE_IMAGE < 0
36 THEN North_Hemisphere = 0;
37 ELSE North_Hemisphere = 1;
38
39 IF DIAM_CIRCLE_IMAGE < 1.53 /* Median */
40 THEN Big_Diameter = 0;
41 ELSE Big_Diameter = 1;
42
43 IF DEPTH_RIMFLOOR_TOPOG <= 0 /* Median */
44 THEN Big_Depth = 0;
45 ELSE Big_Depth = 1;
46
47 LABEL North_Hemisphere = "North Hemisphere" /* Response - Categorical */
48 Big_Diameter = "Big Diameter" /* Explanatory - Bin into Categorical */
49 Big_Depth = "Big Depth"; /* Explanatory - Bin into Categorical */
50 RUN;
51
52 PROC SUMMARY DATA = SAMPLE PRINT ;
53 CLASS North_Hemisphere;
54 TITLE "Mars' Craters - Summary: North Hemisphere - Sample Data";
55 RUN;
56
57 PROC SUMMARY DATA = SAMPLE PRINT ;
58 CLASS Big_Diameter;
59 TITLE "Mars' Craters - Summary: Big Diameter - Sample Data";
60 RUN;
61
62 PROC SUMMARY DATA = SAMPLE PRINT ;
63 CLASS Big_Depth;
64 TITLE "Mars' Craters - Summary: Big Depth - Sample Data";
65 RUN;
66
67 PROC LOGISTIC
68 DATA = SAMPLE
69 PLOTS = ROC;
70 MODEL North_Hemisphere = Big_Diameter;
71 TITLE "Mars' Craters - Logistic - Hemisphere by Diameter - Sample Data";
72 RUN;
73
74 PROC LOGISTIC
75 DATA = SAMPLE
76 PLOTS = ROC;
77 MODEL North_Hemisphere = Big_Diameter Big_Depth;
78 TITLE "Mars' Craters - Logistic - Hemisphere by Diameter and Depth - Sample Data";
79 RUN;
80
81 TITLE ;