MA in English Linguistics Experimental design and statistics

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MA in English LinguisticsExperimental design and
statistics
Sean Wallis Survey of English Usage University
College London s.wallis_at_ucl.ac.uk
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Outline

• 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

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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?

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Examples

• Some example research questions

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Examples

• Some example research questions
• smoking is good for you

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Examples

• Some example research questions
• smoking is good for you
• dropped objects accelerate toward the ground at
  9.8 metres per second squared

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Examples

• Some example research questions
• smoking is good for you
• dropped objects accelerate toward the ground at
  9.8 metres per second squared
• s is a clitic rather than a word

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Examples

• Some example research questions
• smoking is good for you
• dropped objects accelerate toward the ground at
  9.8 metres per second squared
• s is a clitic rather than a word
• the word shall is used less often in recent years

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Examples

• Some example research questions
• smoking is good for you
• dropped objects accelerate toward the ground at
  9.8 metres per second squared
• s is a clitic rather than a word
• 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

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Testable 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?

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Questions 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?

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Questions of choice

• From the speakers 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

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Questions of choice

• From the speakers 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 hearers perspective

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Questions of choice

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

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Questions of choice

• From the speakers 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 hearers perspective
• 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 ?

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Baselines

• 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 (includingll ), when the choice arises

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Baselines

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

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Baselines

• In many cases it is very difficult to identify
  all cases where the choice arises
• e.g. studying modal verbs
• You may need to pick a different baseline
• Be as specific as you can
• words ? VPs ? tensed VPs ? alternating modals

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Baselines

• In many cases it is very difficult to identify
  all cases where the choice arises
• e.g. studying modal verbs
• You may need to pick a different baseline
• Be as specific as you can
• words ? VPs ? tensed VPs ? alternating modals

alternation different words, same meaning
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Baselines

• In many cases it is very difficult to identify
  all cases where the choice arises
• e.g. studying modal verbs
• You may need to pick a different baseline
• Be as specific as you can
• words ? VPs ? tensed VPs ? alternating modals
• Other hypotheses imply different baselines
• Different meanings of the same word
• e.g. uses of very, as a proportion of all cases
  of very
• very N - the very person
• very ADJ - the very tall person
• very ADV - very slightly moving

alternation different words, same meaning

semasiologicalvariation
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Probability

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

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Probability

• 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)

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Probability

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

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Probability

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

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

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Probability

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

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Probability

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

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Probability

• Based on another, even simpler, idea
• probability p x / n
• where
• frequency x (often, f )
• the number of times something actually happens
• the number of hits in a search
• 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)

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

• cases of will

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Probability

• Based on another, even simpler, idea
• probability p x / n
• where
• frequency x (often, f )
• the number of times something actually happens
• the number of hits in a search
• 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)

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

• cases of will

• total will shall

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Probability

• Based on another, even simpler, idea
• probability p x / n
• where
• frequency x (often, f )
• the number of times something actually happens
• the number of hits in a search
• 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)
• 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

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A 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?

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Lets get some data

• Open DCPSE with ICECUP
• FTF query for first person declarative shall
• repeat for will

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Lets 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
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Modal shall vs. will over time

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

1.0
p(shall shall, will)
shall 100
0.8
0.6
0.4
0.2
shall 0
0.0
1955
1960
1965
1970
1975
1980
1985
1990
1995
(Aarts et al., 2013)
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Modal shall vs. will over time

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

1.0
p(shall shall, will)
shall 100
0.8
0.6
0.4
Is shall going up or down?
0.2
shall 0
0.0
1955
1960
1965
1970
1975
1980
1985
1990
1995
(Aarts et al., 2013)
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Is shall going up or down?

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

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Is shall 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 ?

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Is shall 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?

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The sample and the population

• The corpus is a sample

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The sample and the 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

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The sample and the 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

?
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The sample and the 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
• 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

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Basic inferential statistics

• 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?

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Basic inferential statistics

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

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The Binomial distribution

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

F

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

N 1
X
x
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The Binomial distribution

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

F

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

N 4
X
x
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The Binomial distribution

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

F

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

N 8
X
x
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The Binomial distribution

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

F

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

N 12
X
x
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The Binomial distribution

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

F

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

N 16
X
x
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The Binomial distribution

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

F

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

N 20
X
x
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The Binomial distribution

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

F

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

N 26
X
x
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The 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
p
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The Binomial distribution

• 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
p
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The Binomial distribution

• 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
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Binomial ? Normal

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

F
x
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Binomial ? Normal

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

? population mean P
? standard deviationS ? P(1 P) / n
F

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

z . S
z . S
0.5
0.3
0.1
0.7
p
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Binomial ? Normal

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

? population mean P
? standard deviationS ? P(1 P) / n
F
z . S
z . S

• 95 of the curve is within 2 standard deviations
  of the expected mean

• the correct figure is 1.95996!
• the critical value of z for an error level of
  0.05.

2.5
2.5
95
0.5
0.3
0.1
0.7
p
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The single-sample z test...

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

• If yes, p is significantly different from P

F
observation p
z . S
z . S
0.25
0.25
P
0.5
0.3
0.1
0.7
p
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...gives us a confidence interval

• P z . S is the confidence interval for P
• We want to plot the interval about p

F
z . S
z . S
0.25
0.25
P
0.5
0.3
0.1
0.7
p
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...gives us a confidence interval

• P z . S is the confidence interval for P
• We want to plot the interval about p

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...gives us a confidence interval

• The interval about p is called the Wilson score
  interval

observation p

• 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

F
w
w
(Wallis, 2013)
P
0.25
0.25
0.5
0.3
0.1
0.7
p
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Modal shall vs. will over time

• 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

p(shall shall, will)
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Modal shall vs. will over time

• 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

May be input in a 2 x 2 chi-square test
- or you can check Wilson intervals
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Modal shall vs. will over time

• Plotting modal shall/will over time (DCPSE)

• Small amounts of data / year

1.0
p(shall shall, will)
0.8
0.6
0.4
0.2
0.0
1955
1960
1965
1970
1975
1980
1985
1990
1995
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Modal shall vs. will over time

• Plotting modal shall/will over time (DCPSE)

• Small amounts of data / year
• Confidence intervals identify the degree of
  certainty in our results

1.0
p(shall shall, will)
0.8
0.6
0.4
0.2
0.0
1955
1960
1965
1970
1975
1980
1985
1990
1995
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Modal shall vs. will over time

• Plotting modal shall/will over time (DCPSE)

• Small amounts of data / year
• Confidence intervals identify the degree of
  certainty in our results
• Highly skewed p in some cases
• p 0 or 1 (circled)

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Modal shall vs. will over time

• Plotting modal shall/will over time (DCPSE)

• Small amounts of data / year
• Confidence intervals identify the degree of
  certainty in our results
• We can now estimate an approximate downwards curve

(Aarts et al., 2013)
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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?

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Conclusions

• 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 gt 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 2?2 tests or independent-sample z tests to
  compare two observed samples

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References

• 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 203, 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.wordpre
  ss.com