Simple Statistics for Corpus Linguistics Sean Wallis Survey of English Usage University College London [email protected]

Simple Statistics for Simple Statistics for Corpus LinguisticsCorpus Linguistics

Sean WallisSurvey of English Usage

University College London

[email protected]

OutlineOutline

• Numbers…

• A simple research question– do women speak or write more than men

in ICE-GB?– p = proportion = probability

• Another research question– what happens to speakers’ use of modal shall

vs. will over time?– the idea of inferential statistics– plotting confidence intervals

• Concluding remarks

Numbers...Numbers...

• We are used to concepts like these being expressed as numbers:– length (distance, height)– area– volume– temperature – wealth (income, assets)

Numbers...Numbers...

• We are used to concepts like these being expressed as numbers:– length (distance, height)– area– volume– temperature – wealth (income, assets)

• We are going to discuss another concept:– probability

• proportion, percentage

– a simple idea, at the heart of statistics

ProbabilityProbability

• Based on another, even simpler, idea:– probability p = x / n


• Based on another, even simpler, idea:– probability p = x / n – e.g. the probability that the

speaker says will instead of shall



• where– frequency x (often, f )

• the number of times something actually happens• the number of hits in a search

– e.g. the probability that the speaker says will instead of shall





– cases of will






– baseline n is• the number of times something could happen• the number of hits

– in a more general search – in several alternative patterns (‘alternate forms’)

– cases of will








– cases of will

– total: will + shall








• Probability can range from 0 to 1

– e.g. the probability that the speaker says will instead of shall– cases of will

– total: will + shall

What can a corpus tell us?What can a corpus tell us?

• A corpus is a source of knowledge about language:– corpus– introspection/observation/

elicitation– controlled laboratory experiment– computer simulation




}How do these

differ in what they might tell

us?




• A corpus is a sample of language

}How do these


us?


• A corpus is a source of knowledge about language:– corpus– introspection/observation/elicitation– controlled laboratory experiment– computer simulation

• A corpus is a sample of language, varying by:– source (e.g. speech vs. writing, age...)– levels of annotation (e.g. parsing)– size (number of words)– sampling method (random sample?)

}How do these


us?


• A corpus is a source of knowledge about language:– corpus– introspection/observation/elicitation– controlled laboratory experiment– computer simulation

• A corpus is a sample of language, varying by:– source (e.g. speech vs. writing, age...)– levels of annotation (e.g. parsing)– size (number of words)– sampling method (random sample?)

}How do these


us?

How does this affect the types

of knowledg

e we might

obtain?

}

What can a What can a parsedparsed corpus tell corpus tell us?us?• Three kinds of evidence may be found in

a parsed corpus:


a parsed corpus:

Frequency evidence of a particularknown rule, structure or linguistic event - How often?


a parsed corpus:

Frequency evidence of a particularknown rule, structure or linguistic event

Factual evidence of new rules, etc. - How novel?

- How often?

What can a What can a parsedparsed corpus tell corpus tell us?us?• Three kinds of evidence may be found in a

parsed corpus:


Factual evidence of new rules, etc.

Interaction evidence of relationshipsbetween rules, structures and events - Does X affect

Y?

- How novel?

- How often?

What can a What can a parsedparsed corpus tell corpus tell us?us?• Three kinds of evidence may be found in a

parsed corpus:


Factual evidence of new rules, etc.

Interaction evidence of relationshipsbetween rules, structures and events

• Lexical searches may also be made more precise using the grammatical analysis

- Does X affect Y?

- How novel?

- How often?

A simple research questionA simple research question

• Let us consider the following question:

• Do women speak or write more words than men in the ICE-GB corpus?

• What do you think?

• How might we find out?

Lets get some dataLets get some data

• Open ICE-GB with ICECUP– Text Fragment query for words:

• “*+<{~PUNC,~PAUSE}>”• counts every word, excluding pauses

and punctuation




and punctuation

– Variable query:• TEXT CATEGORY = spoken, written




and punctuation


– Variable query:• SPEAKER GENDER = f, m, <unknown>

combine these3 queries}




and punctuation


– Variable query:• SPEAKER GENDER = f, m, <unknown>

F M <unknown> TOTALTOTAL 275,999 667,934 93,355 1,037,288 spoken 174,499 439,741 1,076 615,316 written 101,500 228,193 92,279 421,972

combine these3 queries}

ICE-GB: gender / written-ICE-GB: gender / written-spokenspoken• Proportion of words in each category

spoken/written by women and men– The authors of some texts are unspecified– Some written material may be jointly

authored

– female/male ratio varies slightly

0 0.2 0.4 0.6 0.8 1

TOTAL

spoken

written femalefemale

malemale

p

ICE-GB: gender / written-ICE-GB: gender / written-spokenspoken• Proportion of words in each category

spoken/written by women and men– The authors of some texts are unspecified– Some written material may be jointly

authored

– female/male ratio varies slightly

0 0.2 0.4 0.6 0.8 1

TOTAL

spoken

written femalefemale

malemale

p

pp (female)(female) = words spoken by = words spoken by women /women /

total words (excluding total words (excluding <unknown>)<unknown>)

pp = Probability = Proportion = Probability = Proportion

• We asked ourselves the following question:– Do women speak or write more words

than men in the ICE-GB corpus?– To answer this we looked at the proportion

of words in ICE-GB that are produced by women (out of all words where the gender is known)

pp = Probability = Proportion = Probability = Proportion

• We asked ourselves the following question:– Do women speak or write more words than men in

the ICE-GB corpus?– To answer this we looked at the proportion of words in

ICE-GB that are produced by women (out of all words where the gender is known)

• The proportion of words produced by women can also be thought of as a probability:– What is the probability that, if we were to pick

any random word in ICE-GB (and the gender was known) it would be uttered by a woman?

Another research questionAnother research question

• Let us consider the following question:

• What happens to modal shall vs. will over time in British English?– Does shall increase or decrease?

• What do you think?

• How might we find out?


• Open DCPSE with ICECUP– FTF query for first person declarative shall:

• repeat for will


• Open DCPSE with ICECUP– FTF query for first person declarative shall:

• repeat for will– Corpus Map:

• DATE Do the first set of queries and then drop into Corpus

Map}

Modal Modal shallshall vs. vs. willwill over time over time

• Plotting probability of speaker selecting modal shall out of shall/will over time (DCPSE)

shallshall = 100% = 100%

shallshall = 0% = 0%0.0

0.2

0.4

0.6

0.8

1.0

1955 1960 1965 1970 1975 1980 1985 1990 1995

p(shall | {shall, will})

(Aarts et al. 2013)



0.0

0.2

0.4

0.6

0.8

1.0

1955 1960 1965 1970 1975 1980 1985 1990 1995


(Aarts et al. 2013)





0.0

0.2

0.4

0.6

0.8

1.0

1955 1960 1965 1970 1975 1980 1985 1990 1995


Is shall going up or down?

(Aarts et al. 2013)



Is Is shall shall going up or down? going up or down?

• Whenever we look at change, we must ask ourselves two things:

Is Is shall shall going up or down? going up or down? • Whenever we look at change, we must ask ourselves two things:

What is the change relative to?– Is our observation higher or lower than we might expect?

• In this case we ask • Does shall decrease relative to shall +will ?

Is Is shall shall going up or down? going up or down? • Whenever we look at change, we must ask ourselves two things:



How confident are we in our results?– Is the change big enough to be reproducible?

The idea of a confidence The idea of a confidence intervalinterval• All observations are imprecise

– Randomness is a fact of life– Our abilities are finite:

• to measure accurately or • reliably classify into types

• We need to express caution in citing numbers

• Example (from Levin 2013):– 77.27% of uses of think in 1920s data

have a literal (‘cogitate’) meaning





• Example (from Levin 2013):– 77.27% of uses of think in 1920s data


Really? Not 77.28, or 77.26?





• Example (from Levin 2013):– 77% of uses of think in 1920s data






• Example (from Levin 2013):– 77% of uses of think in 1920s data


Sounds defensible. But how confident can we be in this number?





• Example (from Levin 2013):– 77% (66-86%*) of uses of think in 1920s

data have a literal (‘cogitate’) meaning





• Example (from Levin 2013):– 77% (66-86%*) of uses of think in 1920s

data have a literal (‘cogitate’) meaning

Finally we have a credible range of values - needs a footnote* to explain how it was calculated.

The ‘sample’ and the The ‘sample’ and the ‘population’‘population’• We said that the corpus was a sample


• Previously, we asked about the proportions of male/female words in the corpus (ICE-GB)– We asked questions about the sample– The answers were statements of fact



• Now we are asking about “British English”

?



• Now we are asking about “British English”– We want to draw an inference

• from the sample (in this case, DCPSE)• to the population (similarly-sampled BrE utterances)

– This inference is a best guess– This process is called inferential statistics

Basic inferential Basic inferential statisticsstatistics

• Suppose we carry out an experiment– We toss a coin 10 times and get 5 heads– How confident are we in the results?

• Suppose we repeat the experiment• Will we get the same result again?

Basic inferential Basic inferential statisticsstatistics

• Suppose we carry out an experiment– We toss a coin 10 times and get 5 heads– How confident are we in the results?

• Suppose we repeat the experiment• Will we get the same result again?

• Let’s try…– You should have one coin– Toss it 10 times– Write down how many heads you get– Do you all get the same results?

The Binomial distributionThe Binomial distribution

• Repeated sampling tends to form a Binomial distribution around the expected mean X

F

N = 1

x

531 7 9

• We toss a coin 10 times, and get 5 heads

X



F

N = 4

x

531 7 9

• Due to chance, some samples will have a higher or lower score

X



F

N = 8

x

531 7 9


X



F

N = 12

x

531 7 9


X



F

N = 16

x

531 7 9


X



F

N = 20

x

531 7 9


X



F

N = 26

x

531 7 9


X

The Binomial distributionThe Binomial distribution• It is helpful to express x as the probability of choosing a head, p, with expected mean P

• p = x / n– n = max. number of

possible heads (10)

• Probabilities are inthe range 0 to 1=percentages

(0 to 100%)

F

p

0.50.30.1 0.7 0.9

P


• Take-home point:– A single observation, say x hits (or p as a

proportion of n possible hits) in the corpus, is not guaranteed to be correct ‘in the world’!

• Estimating the confidence you have in your results is essential

F

p

P

0.50.30.1 0.7 0.9

p


• Take-home point:– A single observation, say x hits (or p as a

proportion of n possible hits) in the corpus, is not guaranteed to be correct ‘in the world’!

• Estimating the confidence you have in your results is essential

– We want to makepredictions about future runs of the same experiment

F

p

P

p

0.50.30.1 0.7 0.9

Binomial Binomial Normal Normal

• The Binomial (discrete) distribution is close to the Normal (continuous) distribution

x

F

0.50.30.1 0.7 0.9

The central limit theoremThe central limit theorem

• Any Normal distribution can be defined by only two variables and the Normal function z

z . S z . S

F

– With more data in the experiment, S will be smaller

p0.50.30.1 0.7

population

mean P

standard deviationS = P(1 – P) / n

The central limit theoremThe central limit theorem

• Any Normal distribution can be defined by only two variables and the Normal function z

z . S z . S

F

2.5% 2.5%

population

mean P

– 95% of the curve is within ~2 standard deviations of the expected mean

standard deviationS = P(1 – P) / n

p0.50.30.1 0.7

95%

– the correct figure is 1.95996!

= the critical value of z for an error level of 0.05.

The single-sample The single-sample zz test...test...

• Is an observation p > z standard deviations from the expected (population) mean P?

z . S z . S

F

P2.5% 2.5%

p0.50.30.1 0.7

observation p• If yes, p is

significantly different from P

...gives us a “confidence ...gives us a “confidence interval”interval”• P ± z . S is the confidence interval for P

– We want to plot the interval about p

z . S z . S

F

P

p0.50.30.1 0.7

2.5% 2.5%

...gives us a “confidence ...gives us a “confidence interval”interval”• P ± z . S is the confidence interval for P

– We want to plot the interval about p

w+

F

P2.5% 2.5%

p0.50.30.1 0.7

observation p

w–

95%

...gives us a “confidence ...gives us a “confidence interval”interval”• The interval about p is called the

Wilson score interval

• This interval reflects the Normal interval about P:

• If P is at the upper limit of p,p is at the lower limit of P

(Wallis, 2013)

F

P2.5% 2.5%

p

w+

observation p

w–

0.50.30.1 0.7


• Simple test: – Compare p for

• all LLC texts in DCPSE (1956-77) with• all ICE-GB texts (early 1990s)

– We get the following data

– We may plot the probabilityof shall being selected,with Wilson intervals

LLC ICE-GB totalshall 110 40 150will 78 58 136total 188 98 286

0.0

0.2

0.4

0.6

0.8

1.0

LLC ICE-GB



• Simple test: – Compare p for

• all LLC texts in DCPSE (1956-77) with• all ICE-GB texts (early 1990s)

– We get the following data

– We may plot the probabilityof shall being selected,with Wilson intervals

0.0

0.2

0.4

0.6

0.8

1.0

LLC ICE-GB

p(shall | {shall, will})LLC ICE-GB total

shall 110 40 150will 78 58 136total 188 98 286

May be input in a

2 x 2 chi-square test

- or you can check Wilson intervals

0.0

0.2

0.4

0.6

0.8

1.0

1955 1960 1965 1970 1975 1980 1985 1990 1995



• Plotting modal shall/will over time (DCPSE)

• Small amounts of data / year



0.0

0.2

0.4

0.6

0.8

1.0

1955 1960 1965 1970 1975 1980 1985 1990 1995

p(shall | {shall, will})• Small amounts

of data / year

• Confidence intervals identify the degree of certainty in our results



0.0

0.2

0.4

0.6

0.8

1.0

1955 1960 1965 1970 1975 1980 1985 1990 1995




• Highly skewed p in some cases

– p = 0 or 1 (circled)



0.0

0.2

0.4

0.6

0.8

1.0

1955 1960 1965 1970 1975 1980 1985 1990 1995




• We can now estimate an approximate downwards curve

(Aarts et al. 2013)

Recap Recap • Whenever we look at change, we must ask ourselves two things:



How confident are we in our results?– Is the change big enough to be reproducible?

ConclusionsConclusions

• An observation is not the actual value – Repeating the experiment might get different results

• The basic idea of these methods is – Predict range of future results if experiment was

repeated• ‘Significant’ = effect > 0 (e.g. 19 times out of 20)

• Based on the Binomial distribution– Approximated by Normal distribution – many uses

• Plotting confidence intervals• Use goodness of fit or single-sample z tests to compare

an observation with an expected baseline• Use 22 tests or two independent sample z tests to

compare two observed samples

ReferencesReferences

• Aarts, B., J. Close, G. Leech and S.A. Wallis (eds). The Verb Phrase in English: Investigating recent language change with corpora. Cambridge: CUP.– Aarts, B., Close, J., and Wallis, S.A. 2013. Choices over time:

methodological issues in investigating current change. Chapter 2.– Levin, M. 2013. The progressive in modern American English.

Chapter 8.

• Wallis, S.A. 2013. Binomial confidence intervals and contingency tests. Journal of Quantitative Linguistics 20:3, 178-208.

• Wilson, E.B. 1927. Probable inference, the law of succession, and statistical inference. Journal of the American Statistical Association 22: 209-212.

• NOTE: Statistics papers, more explanation, spreadsheets etc. are published on corp.ling.stats blog: http://corplingstats.wordpress.com

Documents

Simple Statistics for Corpus Linguistics Sean Wallis Survey of English Usage University College London [email protected]