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

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Research on establishing the importance of the many possible causes of

global warming relies on understanding the evolution of greenhouse gasses

in the Earth’s atmosphere. These originate largely from volcanic eruptions,

which release gasses such as CO

, HCl and HF from the interior of

the planet. If these build up in the atmosphere, they can increase the amount

of solar radiation being absorbed, causing temperatures to increase. We are

fortunate that on Earth these gasses react with liquid water to form minerals

such as CaCO

(calcite, or limestone in rock form) and apatite (CaSO

), so

end up getting stored in geological deposits, mostly in the ocean, instead of

heating up our atmosphere. Incidentally, the planet Venus was not so lucky –

it was too close to the Sun for liquid water to be stable on the surface, so all its

greenhouse gasses ended up in its atmosphere, and the surface temperature

there is an inhospitable 460 °C.

A climatologist investigating paleoclimates on Earth decided to examine

the eﬀects of temperature and humidity on the rate of calcite formation, in

an attempt to help predict the storage of carbon dioxide in calcite in

response to global warming at sub-tropical latitudes. They designed an

experiment to examine the amount of calcite precipitation from seawater

as a function of both temperature and humidity (see Ufnar et al., 2008, for a

related example). Identical beakers of carbonate-rich seawater solutions

were placed in six combinations of three temperatures (20, 30 and 40 °C)

and two humidity levels (33 and 66%). There were four beakers in each

treatment, so 24 were used altogether.

This type of design, where there is a treatment for every combination of

the levels of each factor used, is called a “fully orthogonal” design or an

“orthogonal” design (Table 12.1). If one of the treatments was not included

Table 12.1 Example of an orthogonal two-factor design. There are

three levels of Factor A (temperature) and two levels of Factor B

(humidity) with four experimental units (beakers) in each of the six

possible combinations of the 3 × 2 treatment levels.

Temperature (

Humidity (%) 20 30 40

33 4 beakers 4 beakers 4 beakers

66 4 beakers 4 beakers 4 beakers

12.1 Introduction 143

(for example the combination of 33% humidity with 20 °C), then the design

would not be orthogonal.

The results of the experiment can be displayed as a graph of the

means for each of the six combinations (which are often called cell

means), with temperature on the X axis, the weight of calcite precipitate

on the Y axis, and lines joining the three means within each of the two

levels of humidity. (If you wanted you could show humid ity on the

X axis and have lines joining each of the three temperatures, but it is

easier to visualize when the greatest number of treatment levels are on

the X axis.)

Figure 12.1(a) shows a set of cell means where there is no interaction –

the change in humidity from 33 to 66% (or from 66 to 33%) has the same

eﬀect on calcite precipitation at each temperature (in all cases an increase in

humidity increases calcite by about the same amount). Similarly, the eﬀect

of an increase in temperature from 20 °C through to 40 °C (or vice versa) is

the same at each humidity.

In contrast, Figure 12.1(b) shows interaction. A change in humidity from

33 to 66% does not have the same eﬀect on calcite precipitation at each of

the three temperatures, and a change in temperature from 20 °C through to

40 °C does not have has the same eﬀect on calcite precipitation at each

humidity.

That is all interaction is. When there is a c omplete lack of interaction

(e.g. Figure 12.1(a)) the lines joining the treatment means always

run exactly parallel to eac h other (even though both lines move up,

they move up in parallel). In contrast, when there is interaction

(e.g. Figure 12.1(b)) the lines are not always parallel. As the amount of

interaction increases, the lines become less and less parallel and even-

tually the amount of interaction may reach a point where it is considered

signiﬁca nt.

Interaction between two or more factors is often of great interest to

earth scientists. It may be very helpful to know that a response to one

factor is not uniform across the range of a second factor, or that it is

uniform! For example, if you found that the rate of calcite precipitation is

unexpectedly high only when te mperature is high and humidity is low

(Figure 12.1(b)), this synergistic eﬀect would be an important component

of models predicting carbon dioxide storage in rela tion to global

warming.

144 Two-factor analysis of variance

12.2 What does a two-factor ANOVA do?

Here you need to remember that a single-factor ANOVA partitions the total

variation into two components – the variation among groups (treatment

+ error) and the variation within groups (error), and examines whether there

Temperature (°C)

(a)

Wt. of calcite

precipitates

33% humidity

66% humidity

20 30

Tem

erature (°C)

(b)

Wt. of calcite

precipitates

33% humidity

66% humidity

Figure 12.1 Interaction in a two-factor experiment. (a) No interaction

between the two factors temperature and humidity on calcite precipitation.

A change in humidity from 33 to 66% has the same eﬀect on the amount of

calcite precipitation at each of the three temperatures, and a change in

temperature from 20 °C through to 40 °C has the same eﬀect at each humidity.

(b) An interaction between temperature and humidity on calcite precipitation.

A change in humidity from 33 to 66% does not have the same eﬀect on calcite

precipitation at each of the three temperatures, and a change in temperature

from 20 °C through to 40 °C does not have the same eﬀect on calcite

precipitation at each humidity.

12.2 What does a two-factor ANOVA do? 145

is a signiﬁcant eﬀect of treatment by dividing the among groups mean square

by the within groups mean square. This gives an F ratio and probability

that all the treatment means have come from populations with the same

mean.

A two-factor ANOVA works in a similar way, but partitions the total

variation within a set of data into four components: the among group

variation due to (a) Factor A + error, (b) Factor B + error, (c) interaction

+ error and (d) error.

The way the analysis works is a straightforward extension of the concept

developed to explain single-factor ANOVA, and can also be explained

pictorially. For this we will use the simplest case of a two-factor design

with two levels only of each factor, both of which are ﬁxed.

First, here are some examples of the types of outcomes you might get

from a two-fac tor experiment. We ment ioned in Chapter 1 that colored

gemstones are often treated with some combination of

Co irradiation

and heat to improve their appearance and commercial value. To better

understand the e ﬀects of these variables, a gemologist undertook an

experiment using f our di ﬀerent combinations of heat and irradiation,

with four crystals in e ach treatment. Each of t he 16 experimental units

was cut from a large amethyst crystal, so the starting composition and

appearance of each crystal were exa ctly the same. T he crystals were kept at

four combinations of two temperatures and two diﬀerent

Co doses: after

two months the samples were removed and their opacity (degree of trans-

parency) was examined.

Several diﬀerent outcomes are shown in Figure 12.2. The opacity within

each treatment combination with the 10 kGy radiation dose is indicated by

●, while the treatments exposed to the 100 kGy radiation dose are indicated

by ■.

12.3 How does a two-factor ANOVA analyze these data?

This explanation assumes you are familiar with the one already given for a

single-factor ANOVA in Chapter 10. We are using a two-factor experiment

with two levels of each factor, giving four treatment combinations, each of

which contains four replicates. The design is summarized in Table 12.2.

Both factors are ﬁxed – the researcher is only interested in these speciﬁc

temperatures and radiation doses.

146 Two-factor analysis of variance

50 200

Temperature (°C)

(a)

Crystal

opacity

10 kGy and

100 KGy

Co dose

50 200

Temperature (°C)

(b)

Crystal

opacity

100 kGy

Co dose

10 kGy

Co dose

Crystal

opacity

Temperature (°C)

(c)

200

Co dose

10 kGy and

100 kGy

50 200

Temperature (°C)

(d)

Crystal

opacity

100 kGy

Co dose

10 kGy

Co dose

Crystal

opacity

Tem

erature (°C)

(e)

200

100 kGy

Co dose

10 kGy

Co dose

Figure 12.2 Some of the possible outcomes of an orthogonal two-factor

experiment. Only the means for each treatment combination are shown.

(a) No eﬀect of temperature or radiation dose and no interaction. All

treatment means are the same and the lines joining the means within each

radiation dose are also the same. (b) An eﬀect of radiation dose, but no eﬀect of

temperature and no interaction. The two treatment means for 100 kGy

radiation dose are consistently higher than the two for 10 kGy radiation dose.

The two treatment means for 50 °C are consistently greater than the two for

200 °C. (d) An eﬀect of temperature and radiation dose but no interaction.

All treatment means are diﬀerent, but the change in opacity in relation to a

change in radiation dose from 10 to 100 kGy is the same at each temperature

and vice versa. (e) An eﬀect of temperature and radiation dose and some

interaction. The change in opacity between 10 and 100 kGy dose is not the

same at each temperature and vice versa. Note that all lines joining the

treatments within the same radiation dose are parallel except in example

(e), where there is some interaction between temperature and radiation dose.

12.3 How does a two-factor ANOVA analyze these data? 147

To start, think about the opacity of the amethyst crystals that will result

from these treatments. It will be displaced from the grand mean by four

sources of variation – that associated with Factor A plus Factor B plus

interaction plus error. This is called the total variation in the experiment.

Put formally, the position on the Y axis of each replicate in relation to the

grand mean will be determined by the following formula:

Crystal opacity ¼ Factor A þ Factor B þ interaction þ error

(12:1)

Here you may wish to contrast this with the much simpler equation for the

total variation within a single-factor experiment from Chapter 10:

O of tourmaline ¼ treatment þ error (12:2 copied from 10:1)

Just as in the single-factor ANOVA, the variation within a two-factor

experiment can be partitioned into several additive components. These

are shown in Figures 12.3 to 12.6.

First,theﬁnal opacity of each crystal will be displaced from its respective

cell mean by error only. This is estimated in just the same way as for a single-

factor ANOVA and also called the within group variation or error

(Figure 12.3). The distances between each replicate and its cell mean

are squared and added together to give the within group (error) sum of

squares. The sum of squares is divided by the appropriate degrees of freedom

(herethereare3+3+3+3=12)togivethewithingroup(error)meansquare.

Second, each replicate will be displaced from the grand mean by all

sources of variation in the experiment – the eﬀect of Factor A plus

Factor B plus interaction plus error. This is called the total variation in

Table 12.2 The orthogonal design used to explain how two-factor

ANOVA works in Figures 12.3 to 12.6. There are four combinations of

the two temperatures and two radiation doses, with four

experimental units (in this example, amethyst crystals) in each.

Temperature (

Co dose (kGy) 50 200

10 4 crystals 4 crystals

100 4 crystals 4 crystals

148 Two-factor analysis of variance

the experiment. In Figure 12.4 the distance displaced is shown for all

replicates. These distances can be squared and added together to give the

total sum of squares for the experiment. (Again, this is the same as the

procedure for a single factor ANOVA.)

So far, this is the same procedure used to calculate the within group

(error) variance and total variance for a single-factor ANOVA.

(a) The within group variance (Figure 12.3) which is due to error only can

be calculated from the dispersion of the points around each of their

respective cell means.

(b) The total variance (Figure 12.4) will estimate the total variation in the

experiment (the within group (error) variance plus Factor A, Factor B,

plus interaction) and can be calculated from the dispersion of all the

points around the grand mean.

At this stage you still need separate eﬀects for Factor A (temperature

+ error), Factor B (dose + error) and A × B (interaction + error).

Opacity

Grand mean

°C

200 °C

Figure 12.3 The estimation of within group (error) variation in the

experiment on gemstone opacity when amethyst crystals are exposed to four

diﬀerent combinations of temperature and radiation dose. Each crystal is

shown as a symbol: ● = crystals at 10 kGy

Co dose, ■ = crystals at 100 kGy

Co dose. Horizontal lines indicate the grand mean and each cell mean. The

displacement of each replicate from its cell mean (arrows) will be caused by

error only.

12.3 How does a two-factor ANOVA analyze these data? 149

12.4 How does a two-factor ANOVA separate out the eﬀects

of each factor and interaction?

Two-factor ANOVA separates out the eﬀects of each factor and interaction

in a very elegant way. After having done the preliminary calculations in

Figures 12.3 to 12.4, the data are only considered in relation to each of the

two factors. This is done by ﬁrst ignoring the diﬀerent levels within Factor B

and considering the data only in relation to Factor A (temperature), after

which the same is done for Factor B (radiation dose). These procedures are

shown in Figures 12.5 and 12.6 and allow you to calculate separate sums of

squares for temperature + error and also dose + error. They are called the

simple main eﬀects because they examine each factor in isolation from the

other.

First, the levels of radiation dose are ignored and the data treated as though

they are for a single-factor experiment on temperature only. Here, therefore,

you will have eight replicates within each of the two levels of temperature and

you can calculate a mean for each group. These new means, calculated from

all eight replicates within each treatment, will only be displaced from the

Opacity

°C

200

°C

Grand mean

Figure 12.4 The total variation within the experiment on gemstone opacity.

Each crystal is shown as a symbol: ● = crystals at 10 kGy

Co dose, ■ =crystals

at 100 kGy

Co dose. The heavy horizontal line indicates the grand mean. The

four shorter horizontal lines indicate each cell mean. The displacement of each

replicate from the grand mean (arrows) will be caused by the total variation

within the experiment.

150 Two-factor analysis of variance

grand mean by the average eﬀect of temperature plus error.Therefore,the

displacement of the treatment means from the grand mean can be used to

calculate the sum of squares and mean square for Factor A (temperature +

error) only (Figure 12.5) just as in a single-factor ANOVA.

Second, the levels of temperature are ignored and the data are treated as

though they are for a single-factor experiment on radiation dose only.

Here too, you will have eight replicates within each of the two levels of

radiation dose and you can calculate a mean for each of the two g roups.

These new means, calculated from all eight replicates within each treat-

ment, will only be displaced from the grand m ean by the average eﬀect

of dose plus error (Figure 12.6). Therefore, the displacement of the

treatment means from the grand mean can be used to calculate the sum

of squares and mean square for Factor B (radiation dose + e rror) only just

as in a s ingle- fact or A NOVA.

At this stage you have sums of squares for the following:

Opacity

Grand mean

°C

200

°C

Figure 12.5 The eﬀect of Factor A (temperature + error) only on the opacity

of amethyst crystals. Each crystal is represented as a symbol: ● = crystals at

10 kGy

Co dose, ■ = crystals at 100 kGy

Co dose. These data have been

pooled for each temperature, ignoring radiation dose, thereby generating two

new treatment means, shown by the horizontal lines. The displacement of

each treatment mean from the grand mean is an estimate of the average eﬀect

of temperature plus error. The sum of squares is the sum of the square of each

displacement, which is then multiplied by the number of replicates in that

treatment. The mean square is the sum of squares divided by n − 1 degrees of

freedom where n is the number of temperature treatments (here n = 2).

12.4 Eﬀects of each factor and the interaction 151

(a) The total variation in the experiment (the combined eﬀects of Factor

A, Factor B, A × B and error) (Figure 12.4)

(b) The eﬀect of Factor A (temperature + error) (Figure 12.5)

(d) error (Figure 12.3)

From this list, the only separate sum of squares you still need is the one

for interaction plus error. Because the sums of squares are additive and the

total variation is the combined eﬀects of all the factors in the ANOVA

(Section 10.3.4), you can calculate the sum of squares for interaction by

subtraction. This is done by taking away the sums of squares for Factor A,

Factor B, and error from the total sum of squares ((a) above minus (b) and

The total variation in the experiment (the combined eﬀects of Factor A,

Factor B, A × B and error) (Figure 12.4)

The eﬀect of Factor A (temperature + error) (Figure 12.5)

Opacity

Grand mea

10 kGy

100 kG

Figure 12.6 The eﬀect of Factor B (radiation dose + error) only on the

opacity of amethyst crystals, represented here as symbols: ● = crystals at

10 kGy

Co dose, ■ = crystals at 100 kGy

Co dose. These data have been

pooled for each radiation dose, ignoring temperature, thereby generating two

diﬀerent treatment means, shown by the horizontal lines. The displacement of

each treatment mean from the grand mean is the average eﬀect of dosage for

the number of replicates in that treatment. The sum of squares for the eﬀ ect of

radiation dose is the sum of the square of each displacement, which is then

multiplied by the number of replicates in that treatment. The mean square is

the sum of squares divided by n − 1 degrees of freedom where n is the number

of radiation treatments (here n = 2).

152 Two-factor analysis of variance