Steve M., Darby D.M., Geostatistics Explained - An Introductory Guide for Earth Scientists

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isthevarianceandthevariancetomeanratioofthenumberofminesper

quadrat?(c)Isthisratiosigniﬁcantly diﬀerent from 1.0? (d) Is the distri-

bution of silver mines per quadrat more or less uniform than random?

(2) Behavior patterns of dinosaurs may be recorded in the geologic record in

serendipitous circumstances, such as the many sets of tracks exposed in

what is now the Connecticut River Valley in the eastern US. A paleon-

tologist wanted to test the idea that an exposure of 32 diﬀerent dinosaur

footprints in a large slab of sandstone show the movements of a herd,

pack, or ﬂock of dinosaurs all moving in the same direction. The follow-

ing data are for the direction, in degrees, of 32 fossil footprints: 101°, 134°,

205°, 95°, 164°, 199°, 223°, 144°, 173°, 243°, 105°, 164°, 266°, 119°, 146°,

169°, 94°, 155°, 250°, 144°, 170°, 201°, 215°, 99°, 134°, 219°, 232°, 140°,

178°, 232°, 107° and 161°. (a) Use these data to plot a rose diagram, where

petal area is proportional to frequency, showing the direction of the

footprints. (b) By inspection of this diagram, do the directions of the

footprints appear to be evenly distributed? (c) Test this hypothesis.

(3) A homeowner decided to plant a line of trees along the edge of her lot to

give some shelter from winds and storms. She needed to know the most

frequent direction (if any) of the wind in order to maximize the

protection provided by the wind break. The following data are for the

direction, in degrees, of weekly average wind directions recorded over

the course of a year at a nearby airport: 2°, 164°, 272°, 82°, 330°, 200°,

30°, 164°,123°,180°, 224°, 302°, 44°, 12°, 294°, 316°, 205°, 164°,150°,

242°, 107°, 64°, 33°, 348°, 12°, 124°, 242°, 182°, 350°, 241°, 50°,

134°,163°, 70°, 204°, 332°, 144°, 112°, 94°, 16°, 305°, 264°, 50°, 142°,

127°, 164°, 233°, 48°, 154°, 293°, 210° and 48°. (a) Use these data to plot

a rose diagram, where petal area is proportional to frequency, showing

the direction of the winds. (b) By inspection of this diagram do the wind

directions appear to be evenly distributed? (c) Test this hypothesis.

(4) What is the diﬀerence between the theoretical and the empirical

semivariogram?

(5) Comment on the statement “If there is no regional dependence the

semivariogram has little or no use in interpolation such as Kriging.”

22.6 Questions 363

23 Choosing a test

23.1 Introduction

Statisticians and earth scientists who teach statistics are often visited in their

oﬃces by a researcher or student they may never even have met before, who

is clutching a dauntingly thick pile of paper and perhaps a couple of ﬂash

drives or CDs with labels like “Experiment 1” or “Trial 2”. The visitor sits

down, drops everything heavily on the desk and says, “Here are my results.

What stats do I need?”

This is not a good thing to do. First, the person whose advice you are

asking may not have the time to work out exactly what you have done, so

they may give you bad advice. Second, the answer can be a very nasty

surprise like “There are problems with your experimental design.”

The decision about the appropriate statistical analysis needs to be made

by considering the hypothesis being tested, the experimental design and the

type of data. It can save a lot of time, trouble and disappointment if you

think about possible ways of analyzing the data before the sampling is done

or the experiment designed, rather than only after the data have been

collected.

Tables 23.1–23.12 are a guide to choosing an appropriate analysis from

the ones discussed in this book. You need to start at Table 23.1, which

initially gives ﬁve columns that are mutually exclusive alternatives. Once

you have decided among these, work downwards within the column you

have chosen. There may be more choices and again you need to select the

appropriate column and continue downwards. Eventually you will be

referred to another table with more choices that lead to suggested analyses.

364

Table 23.1 Are you: (a) testing the hypothesis that one or more samples of univariate, ratio, interval ordinal or nominal scale data are

from the same population, (b) testing whether two variables are related, (c) examining whether there is similarity or dissimilarity

among samples of multivariate data, (d) analyzing and/or modeling a temporal or spatial sequence, or (e) analyzing the spatial

properties of a distribution of points in a two-dimensional sampling space?

(a) Testing the hypothesis that one or more univariate samples of ratio,

interval, ordinal or nominal scale data are from the same population

For ratio, interval or ordinal scale

data

For example:

Comparing the mean of a sample with

an expected value of 0.025 wt%:

wt% gold

0.021

0.014

0.016

0.007

0.016

0.010

or comparing the means of two or

more samples (e.g. length of clam

fossils at three different outcrops):

Site J Site K Site L

_____ ____ ____

6 13 28

3 17 26

1 20 23

4 16 26

_____________________

Go to Table 23.2

For nominal scale data

Nominal scale data are numbers in two

or more mutually exclusive categories

such as the number of accumulating

vs. ablating glaciers (two categories)

for a sample of 200:

or the numbers in two or more mutually

exclusive categories (e.g. accumulation

vs. ablation at several different places

in the Alps):

Go to Table 23.4

Accumulation Ablation

107 93

Site

Accumulation 32 21 36

Ablation 14 23 17

(b) Testing whether two

variables are related

Data must be bivariate:

measured on a ratio, interval or

ordinal scale and can be

displayed as a two-dimensional

scatter plot.

Go to Table 23.9

Case X Y

1 2 7

2 4 11

3 6 13

4 8 15

5 10 18

6 12 19

7 14 22

8 16 23

whether there is

similarity or

dissimilarity

among samples

of multivariate

data

Go to Table

23.10

(d) Analyzing

and/or modeling

a temporal or

spatial sequence

Go to Table

23.11

(e) Analyzing

the spatial

properties of a

distribution of

points in a two-

dimensional

sampling space

Go to Table

23.12

Table 23.2 Is the hypothesis about: (a) whether one sample is from a population with a known distribution and thus known population

statistics, (b) whether two or more samples are from the same population?

One sample of ratio,

interval or ordinal

scale data is from a

population with known

(or expected) statistics

Two or more samples of ratio, interval or ordinal scale data are from the same population

Are the samples independent?

Case Sample

Sample

1 X

2 X

3 X

4 X

5 X

6 X

Are the samples related?

Case Sample

Sample

1 X X X

2 X X X

3 X X X

4 X X X

5 X X X

6 X X X

Go to Table 23.3

Case Data

1 165

2 150

3 159

4 145

5 170

Can you assume the data are

normally distributed?

Go to Table 23.5

Are the data grossly non-normal?

Go to Table 23.6

Can you assume the data are

normally distributed?

Got to Table 23.7

Are the data grossly non-normal?

Go to Table 23.8

Table 23.3 Tests for one sample of ratio, interval or ordinal scale data.

Is the sample from a

population that appears

normal, or not grossly non-

normal?

Z test if the population mean

and variance are known

(Chapter 8) but single-sample t

test if there is only an expected

value for the population mean

(Chapter 8)

Is the sample from a

population that is grossly

non-normal (e.g. bimodal?)

Kolmogorov–Smirnov one-

sample test (Chapter 19)

23.1 Introduction 367

Table 23.4 Tests for nominal scale data.

To test whether one

sample of nominal

scale data is from a

population with

known statistics

To test whether two or more samples of nominal scale data are from the same population

Are the samples independent? Are the samples related?

Are the frequencies in two

or more mutually

exclusive categories and

you want to compare

them to an expected ratio?

(For example, comparison

of the sex of 110

Ptyctodon fossils to an

expected ratio of 1:1).

Non-parametric

goodness-of-fit test

(Chapter 18)

Male 54

Female 56

Can the data be cast as a

table?

Sample

Male 12 33

Female 42 27

A Fisher Exact Test or a chi-square

test for a

2 × 2 contingency table

(Chapter 18)

Caution: For small sample sizes a

Fisher Exact Test is recommended

Can the data be cast as a table with two

or more columns and rows? (For

example, the numbers of meteorites in

the three mutually exclusive categories

of stone, iron and stony iron, in three

different samples).

Sample

Stone 23 14 24

Iron 12 19 14

Stony

iron

15 12 13

A chi-square test, exact test or

randomization test for a contingency

table (Chapter 18)

Caution: Exact tests are recommended

when samples sizes are small

Are there two samples?

A McNemar test for the

significance of changes or an

exact or randomization test

(Chapter 18)

Case Before

treatment

After

treatment

1 Yes No

2 Yes No

3 No No

4 Yes Yes

5 No No

6 Yes No

Are there more than two samples?

Case Condition

Condition

1 Yes No No

2 Yes No Yes

3 No No No

4 Yes Yes Yes

5 No No Yes

6 Yes No No

A Cochran Q test or an exact or

randomization test (Chapter 18)

Table 23.5 Tests for two or more independent samples of normally distributed ratio, interval or ordinal scale data.

To test whether two or more independent samples of normally distributed ratio, interval or ordinal scale data are from the same population

Are the data replicates for two or more levels of one factor (e.g.

Rb content of three different size fractions)?

<45 µm 45–125 µm 125–250 µm

_______ _______ _______

6 13 28

3 17 26

1 20 23

4 16 26

Two samples Three or more samples

Are the data an orthogonal design for two or more levels of two

factors (e.g. precipitated crystal mass in relation to temperature

and cooling rate)?

_______________ _______________ _____________

Slow Fast Slow Fast Slow Fast

cooling cooling cooling cooling cooling cooling

____ ____ ____ ____ ____ ____

6 1 13 3 28 5

3 3 17 5 26 7

1 2 20 2 23 3

4 1 16 5 26 4

Data are replicated within each combination of

treatments

Sample

variances similar

Independent

sample t test

(Chapter 8) or

single-factor

ANOVA

(Chapter 10)

Sample

variances

grossly

dissimilar

Independent

sample t test

(Chapter 8)

which does not

assume equal

variances

Sample

variances

similar

Single-

factor

ANOVA

(Chapter

10)

Sample

variances

grossly

dissimilar

Transform

and recheck.

If similar,

use single-

factor

ANOVA

(Chapter

10). If not,

analyze with

caution

because of

the

increased

risk of Type

1 error

(Chapter 13)

Variances are

similar among

treatment

combinations

Two-factor

ANOVA (Chapter

12)

Variances are grossly

dissimilar among treatment

combinations

Transform and recheck

variances. If similar, use

two-factor ANOVA

(Chapter 12). Otherwise,

use ANOVA but be aware

of the increased risk of Type

1 error (Chapter 13)

Data are not

replicated

Two-factor

ANOVA

without

replication

(Chapter 14)

Are the data an

orthogonal

design for two

or more levels

of three or

more factors

(e.g. stream

discharge in

relation to

slope, velocity

and

vegetation)?

Three-factor

OVA

subject to the

cautions and

conditions for

two-factor

ANOVA in the

columns to the

left of this one

Are the data a nested design where one

or more factors are nested within the

levels of another? (For example, the

effect of temperature on crystal

preci

pitation, with two growth

chambers nested within each

temperature)

_________ __________ _________

Chamber Chamber Chamber

A B C D E F

___ ___ ___ ___ ___ ___

6 5 13 15 28 34

3 6 17 19 26 19

2 1 20 13 23 26

1 1 16 17 26 28

Nested ANOVA (Chapter 14), subject

to the cautions about inequality of

variances noted in the columns for

single-factor and two-factor ANOVA to

the left of this one

20 °C 30 °C 40 °C

40 °C30 °C20 °C

Table 23.6 Tests for two or more independent samples of ratio, interval or ordinal

scale data that are not normally distributed.

To test whether two or more independent samples of ratio, interval or ordinal

scale data, that are grossly non-normal, are from the same population

Are the data for two levels or samples

of one factor (e.g. garnet in rocks from

two diﬀerent metamorphic grades, or

zircon content from two outcrops)?

Are the data for three or more levels or

samples of one factor (e.g. quartz content of

sediments of three sizes, or trace fossils at

three or more outcrops)?

Outcrop 1 Outcrop 2 boulders cobbles pebbles

6 13 6 13 28

3 17 3 17 26

1 20 1 20 23

4 16 4 16 26

Sample

distributions

similar

Sample

distributions

grossly diﬀerent

Sample distributions

similar

Sample

distributions

grossly diﬀerent

Mann–

Whitney U test,

randomization

test or exact test

(Chapter 19)

Transform to a

nominal scale

and analyze as

categorical

data (Chapter 19)

Kruskal–Wallis test,

randomization test, or

exact test (Chapter 19)

Transf orm to a

nominal scale

and analyze as

categorical data

(Chapter 19)

370 Choosing a test

Table 23.7 Tests for two or more related samples of normally distributed ratio,

interval or ordinal scale data.

To test whether two or more related samples of normally distributed data are

from the same population

Two related samples Three or more related samples

Case Sample 1 Sample 2 Case Sample 1 Sample 2 Sample 3

1 12 15 1 12 15 23

2 16 12 2 16 12 18

3 21 17 3 21 17 26

4 18 10 4 18 10 21

5 19 14 5 19 14 29

6 12 18 6 12 18 24

Paired-sample t test (Chapter 8)or

two-factor ANOVA without

replication (Chapter 14)

Two-factor ANOVA without replication

(Chapter 14)

Table 23.8 Tests for two or more related samples of ratio, interval or ordinal scale

data that are not normally distributed.

To test whether two or more related samples of data, that are grossly non-normal,

are from the same population

Two related samples Three or more related samples

Case Sample 1 Sample 2 Case Sample 1 Sample 2 Sample 3

1 12 15 1 12 15 23

2 16 12 2 16 12 18

3 21 17 3 21 17 26

4 18 10 4 18 10 21

5 19 14 5 19 14 29

6 12 18 6 12 18 24

Wilcoxon paired-sample test,

randomization test or exact test

(Chapter 19)

Friedman test, randomization test or exact

test (Chapter 19)

23.1 Introduction 371

Table 23.9 Tests for whether two variables are related.

To test whether two variables are related

Do X and Y vary together?

Do the data fit the

bivariate normal

distribution and does the

relationship appear linear?

Linear correlation

analysis (Chapter 15)

Are the data grossly non-

normal or is the relationship

non-linear?

Non-parametric

correlation analysis

(Chapter 19)

What is the functional relationship between X

and Y ?

Y = a + bX

Linear regression analysis (Chapter 16)

Cautions: Data must be normally distributed and

the relationship must be linear. Transformation

of the Y variable may be needed

Table 23.10 Methods for comparing two or more samples for which multivariate

data are available.

To isolate variables or combinations

of variables that can help explain any

pattern of similarity or dissimilarity

among a group of multivariate

samples

To display multivariate samples in a

reduced two-dimensional space in

order to give an easily interpreted

summary of their similarities and

dissimilarities

Principal components analysis

(Chapter 20)

Display as points in

two-dimensional space

Multidimensional

scaling (Chapter 20)

Display as a

dendogram

Cluster

analysis

(Chapter 20)

372 Choosing a test