MA in English Linguistics Experimental design and statistics Sean Wallis Survey of English Usage University College London [email protected]

MA in English LinguisticsMA in English LinguisticsExperimental design and statisticsExperimental design and statistics

Sean WallisSurvey of English Usage

University College London

[email protected]

OutlineOutline

• What is a research question?

• Choice and baselines

• Making sense of probability

• Observing change in a corpus

• Drawing inferences to larger populations

• Estimating error in observations

• Testing results for significance

What is a research question?What is a research question?

• You may have heard this phrase last term

• What do you think we mean by a “research question”?

• Can you think of any examples?

ExamplesExamples

• Some example research questions

ExamplesExamples


– smoking is good for you

ExamplesExamples



– dropped objects accelerate toward the ground at 9.8 metres per second squared

ExamplesExamples




– ’s is a clitic rather than a word

ExamplesExamples





– the word shall is used less often in recent

years

ExamplesExamples





– the word shall is used less often in recent years

– the degree of preference for shall rather than will has declined in British English over the period 1960s-1990s

Testable hypothesesTestable hypotheses

• An hypothesis = a testable research question

• Compare– the word shall is used less in recent years

to– the degree of preference for shall rather

than will has declined in British English over the period 1960s-1990s

• How could you test these hypotheses?

Questions of choiceQuestions of choice

• Suppose we wanted to test the following hypothesis using DCPSE

– the word shall is used less in recent years

• When we say the word shall is used less...

– ...less compared to what?• traditionally corpus linguists have “normalised” data as

a proportion of words (so we might say shall is used less frequently per million words)

• But what might this mean?


• From the speaker’s perspective:– The probability of a speaker using a word like shall

depends on whether they had the opportunity to say it in the first place

– They were about to say will, but said shall instead





– Per million words might still be relevant from the hearer’s perspective






• If we can identify all points where the choice arose, we have an ideal baseline for studying linguistic choices made by speakers/writers.


• From the speaker’s perspective:– The probability of a speaker using a word like shall depends

on whether they had the opportunity to say it in the first place



• If we can identify all points where the choice arose, we have an ideal baseline for studying linguistic choices made by speakers/writers.

– Can all cases of will be replaced by shall ?– What about second or third person shall ?

BaselinesBaselines

• The baseline is a central element of the hypothesis– Changes are always relative to something– You can get different results with different baselines– Different baselines imply different conclusions

• We have seen two different kinds of baselines– A word baseline

• shall per million words– A choice baseline (an “alternation experiment”)

• shall as a proportion of the choice shall vs. will (including’ll ), when the choice arises

BaselinesBaselines

• In many cases it is very difficult to identify all cases where “the choice” arises– e.g. studying modal verbs

BaselinesBaselines


• You may need to pick a different baseline– Be as specific as you can

• words VPs tensed VPs alternating modals

BaselinesBaselines




alternation = “different

words, same meaning”

BaselinesBaselines




• Other hypotheses imply different baselines:– Different meanings of the same word:

• e.g. uses of very, as a proportion of all cases of veryvery +N - the very personvery +ADJ - the very tall personvery +ADV - very slightly moving

alternation = “different

words, same meaning”

semasiological

variation}

ProbabilityProbability

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


• 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)


• 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

A simple research questionA simple research 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?

Lets get some dataLets get some data

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

• repeat for will

Lets get some dataLets get some data

• 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)

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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)

shallshall = 100% = 100%



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

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1955 1960 1965 1970 1975 1980 1985 1990 1995


Is shall going up or down?




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?– What is our baseline for comparison? – 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:

What is the change relative to?– What is our baseline for comparison? – In this case we ask

• Does shall decrease relative to shall +will ?

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

The ‘sample’ and the The ‘sample’ and the ‘population’‘population’• The corpus is a sample


• If we ask questions about the proportions of certain words in the corpus– We ask questions about the sample– Answers are 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


• 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


• 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

P0.25% 0.25%

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

P0.25% 0.25%

p0.50.30.1 0.7

...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

P0.25% 0.25%

p0.50.30.1 0.7

observation p

w–

...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

P0.25% 0.25%

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

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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

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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

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1955 1960 1965 1970 1975 1980 1985 1990 1995



• Plotting modal shall/will over time (DCPSE)

• Small amounts of data / year



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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



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1955 1960 1965 1970 1975 1980 1985 1990 1995




• Highly skewed p in some cases

– p = 0 or 1 (circled)



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1955 1960 1965 1970 1975 1980 1985 1990 1995




• We can now estimate an approximate downwards curve


Recap Recap • 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 ?

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 inferential statistics 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 independent-sample z tests to compare

two observed samples

ReferencesReferences

• Aarts, B., Close, J., and Wallis, S.A. 2013. Choices over time: methodological issues in investigating current change. Chapter 2 in Aarts, B. Close, J., Leech G., and Wallis, S.A. (eds.) The Verb Phrase in English. Cambridge University Press.

• 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

MA in English Linguistics Experimental design and statistics Sean Wallis Survey of English Usage University College London [email protected]