Week 4. Statistics etc.Week 4. Statistics etc.

GRS LX 865GRS LX 865Topics in Topics in

LinguisticsLinguistics

Inversion in negationInversion in negation

Guasti, Thornton & Wexler (BUCLD Guasti, Thornton & Wexler (BUCLD 1995) looked at doubling in negative 1995) looked at doubling in negative questions.questions.

Previous results (Bellugi 1967, 1971, Previous results (Bellugi 1967, 1971, Stromswold 1990) indicated that kids Stromswold 1990) indicated that kids tend to invert less often in negative tend to invert less often in negative questions.questions. First: True?First: True? Second: Why?Second: Why?

GTW (1995)GTW (1995)

Elicited negative questions…Elicited negative questions… I heard the snail doesn’t like some things I heard the snail doesn’t like some things

to eat. Ask him what.to eat. Ask him what. There was one place Gummi Bear There was one place Gummi Bear

couldn’t eat the raisin. Ask the snail couldn’t eat the raisin. Ask the snail where.where.

One of these guys doesn’t like cheese. One of these guys doesn’t like cheese. Ask the snail who.Ask the snail who.

I heard that the snail doesn’t like potato I heard that the snail doesn’t like potato chips. Could you chips. Could you ask ask him if he doesn’t?him if he doesn’t?

GTW (1995)GTW (1995)

Kids got positive questions rightKids got positive questions right for for the most part.the most part. 88% of kids’ wh-questions had inversion88% of kids’ wh-questions had inversion 96% of kids’ yes-no questions had 96% of kids’ yes-no questions had

inversioninversion Except youngest kid (3;8), who had Except youngest kid (3;8), who had

inversion only 42% of the time.inversion only 42% of the time. Kids got negative declaratives right Kids got negative declaratives right

without exception, with without exception, with dodo-support -support and clitic and clitic n’tn’t..

GTW (1995)GTW (1995) Kids got lots of negative Kids got lots of negative whwh-questions -questions wrongwrong.. Aux-doublingAux-doubling

What kind of bread do you don’t like? (3;10)What kind of bread do you don’t like? (3;10) Neg & Aux doublingNeg & Aux doubling

Why can’t she can’t go underneath? (4;0)Why can’t she can’t go underneath? (4;0) No I to C raising (inversion)No I to C raising (inversion)

Where he couldn’t eat the raisins? (4;0)Where he couldn’t eat the raisins? (4;0) Not Not structurestructure

Why can you not eat chocolate? (4;1)Why can you not eat chocolate? (4;1)

GTW (1995)GTW (1995) But kids got negative But kids got negative subjectsubject whwh-questions -questions

right.right. which one doesn’t like his hair messed up? (4;0)which one doesn’t like his hair messed up? (4;0)

……as well as as well as how-come how-come questions.questions. How come the dentist can’t brush all the teeth? How come the dentist can’t brush all the teeth?

(4;2)(4;2)

Re: Re: Not Not structurestructure Why can you not eat chocolate? (4;1)Why can you not eat chocolate? (4;1) Kids only do this with object and adjunct Kids only do this with object and adjunct whwh--

questionsquestions—if kids just sometimes prefer —if kids just sometimes prefer notnot instead of instead of n’tn’t, we would expect them to use it just , we would expect them to use it just as often with subject as often with subject whwh-questions.-questions.

GTW (1995)GTW (1995) So, in sum:So, in sum:

Kids get positive questions rightKids get positive questions right Kids get negative declaratives rightKids get negative declaratives right Kids get negative subject questions right.Kids get negative subject questions right. Kids get negative Kids get negative how-come how-come questions right.questions right.

Kids make errors in negative Kids make errors in negative whwh--questions where questions where inversioninversion is is requiredrequired. Where inversion isn’t required . Where inversion isn’t required (or where the sentence isn’t negative), (or where the sentence isn’t negative), they’re fine.they’re fine.

GTW (1995)GTW (1995) The kids’ errors all seem to have the The kids’ errors all seem to have the

character of character of keeping negation inside the IPkeeping negation inside the IP.. What did he didn’t wanna bring to school? (4;1)What did he didn’t wanna bring to school? (4;1) What she doesn’t want for her witch’s brew? (3;8)What she doesn’t want for her witch’s brew? (3;8) Why can you not eat chocolate? (4;1)Why can you not eat chocolate? (4;1) Why can’t she can’t go underneath? (4;3)Why can’t she can’t go underneath? (4;3)

GTW propose that this is a legitimate option; GTW propose that this is a legitimate option; citing Paduan (Italian dialect) as a language citing Paduan (Italian dialect) as a language doesn’t allow neg->C.doesn’t allow neg->C.

GTW (1995)GTW (1995) Re: subject and Re: subject and how come how come questions…questions… In a subject question, we don’t In a subject question, we don’t knowknow that that

the subject the subject whwh-word got out of IP—maybe -word got out of IP—maybe kids left it in IP… heck, maybe even kids left it in IP… heck, maybe even adultsadults do.do. Who left?Who left? *Who did leave?*Who did leave?

How comeHow come questions don’t require SAI in questions don’t require SAI in the adult language{./?}the adult language{./?} How come John left?How come John left? *How come did John leave?*How come did John leave?

Descriptive, inferentialDescriptive, inferential Any discussion of statistics anywhere (± a Any discussion of statistics anywhere (± a

couple) seems to begin with the following couple) seems to begin with the following distinction:distinction:

Descriptive statisticsDescriptive statistics Various measures used to describe/summarize Various measures used to describe/summarize

an existing set of data. Average, spread, … an existing set of data. Average, spread, … Inferential statisticsInferential statistics

Similar-looking measures, but aiming at Similar-looking measures, but aiming at drawing conclusions about a population by drawing conclusions about a population by examining a sample.examining a sample.

Central tendency and Central tendency and dispersiondispersion

A good way to summarize a set of A good way to summarize a set of numbers (e.g., reaction times, test numbers (e.g., reaction times, test scores, heights) is to ascertain a “usual scores, heights) is to ascertain a “usual value” given the set, as well as some value” given the set, as well as some idea of how far values tend to vary from idea of how far values tend to vary from the usual.the usual.

Central tendency:Central tendency: mean (average), median, modemean (average), median, mode

Dispersion:Dispersion: Range, variance (Range, variance (SS22), standard deviation (), standard deviation (SS))

Data points relative to Data points relative to the distribution: the distribution: zz-scores-scores Once we have the summary Once we have the summary

characteristics of a data set (mean, characteristics of a data set (mean, standard deviation), we can describe any standard deviation), we can describe any given data point in terms of its position given data point in terms of its position relative to the mean and the distribution relative to the mean and the distribution using a standardized score (the using a standardized score (the zz-score).-score).

The The zz-score is defined so that 0 is at the -score is defined so that 0 is at the mean, -1 is one standard deviation below, mean, -1 is one standard deviation below, and 1 is one standard deviation above:and 1 is one standard deviation above:

€

zi =x i −M

S

Type I and Type II Type I and Type II errorserrors

As a reminder, as As a reminder, as we evaluate data we evaluate data sampled from the sampled from the world to draw world to draw conclusions, there conclusions, there are four are four possibilities for any possibilities for any given hypothesis:given hypothesis: The hypothesis is The hypothesis is

(in reality) either (in reality) either true or falsetrue or false

We conclude that We conclude that the hypothesis is the hypothesis is true or false.true or false.

Inno-Inno-centcent

GuiltyGuilty

ConvicConvictt

Type I Type I errorerror

CorrecCorrectt

AcquitAcquit CorrecCorrectt

Type Type II II

errorerrorThis leaves two outcomes thatare correct, and two that areerrors.

Type I and Type II Type I and Type II errorserrors

The risk of making a The risk of making a Type I error is Type I error is counterbalanced by the counterbalanced by the risk of making Type II risk of making Type II errors; being safer with errors; being safer with respect to one means respect to one means being riskier with being riskier with respect to the other.respect to the other.

One needs to decide One needs to decide which is worse, what which is worse, what the acceptable level of the acceptable level of risk is for a Type I risk is for a Type I error, and establish a error, and establish a criterioncriterion— a threshold — a threshold of evidence that is of evidence that is needed in order to needed in order to decide to convict.decide to convict.

Inno-Inno-centcent

GuiltyGuilty

ConvicConvictt

Type I Type I errorerror

CorrecCorrectt

AcquitAcquit CorrecCorrectt

Type Type II II

errorerrorYou may sometimes encounterType I errors referred to as errors,and Type II errors as errors.

Binomial/sign testsBinomial/sign tests If you have an If you have an

experiment in experiment in which each which each trialtrial has two possible has two possible outcomes (coin flip, outcomes (coin flip, rolling a 3 on a die, rolling a 3 on a die, kid picking the kid picking the right animal out of right animal out of 6), you can do a 6), you can do a binomial testbinomial test.. Called a Called a sign testsign test if if

success and failure success and failure have equal have equal probabilities (e.g. probabilities (e.g. coin toss)coin toss)

Hsu & Hsu’s (1996) example: Hsu & Hsu’s (1996) example: Kid asked to pick an animal in Kid asked to pick an animal in response to stimulus sentence. response to stimulus sentence. Picking the right animal (of 6) Picking the right animal (of 6) serves as evidence of knowing serves as evidence of knowing the linguistic phenomenon the linguistic phenomenon under investigation.under investigation.

Random choice would yield 1 Random choice would yield 1 out of 6 chance out of 6 chance (probability .17) of getting it (probability .17) of getting it right. Success.right. Success. Failure: probability 1-.17=.83Failure: probability 1-.17=.83

Chances of getting it right 4 Chances of getting it right 4 times out of 5 by guessing times out of 5 by guessing = .0035. Chances of getting it = .0035. Chances of getting it right all 5 times is .0001.right all 5 times is .0001.

Hypothesis testingHypothesis testing Independent variableIndependent variable

is one which we control.is one which we control. Dependent variableDependent variable is is

the one which we the one which we measure, and which we measure, and which we hypothesize may be hypothesize may be affected by the choice of affected by the choice of independent variable.independent variable.

Summary scoreSummary score: What : What we’re measuring about we’re measuring about the dependent variable. the dependent variable. Perhaps number of Perhaps number of times a kid picks the times a kid picks the right animal.right animal.

HH00: The independent : The independent variable has no effect on variable has no effect on the dependent variable.the dependent variable. A grammatically indicated A grammatically indicated

animal is not more likely animal is not more likely to be picked.to be picked.

HH11: The independent : The independent variable variable doesdoes have an have an effect on the dependent effect on the dependent variable.variable. A grammatically indicated A grammatically indicated

animal is more likely to be animal is more likely to be picked.picked.

Hypothesis testingHypothesis testing HH00: The independent : The independent

variable has no effect variable has no effect on the dependent on the dependent variable.variable. A grammatically A grammatically

indicated animal is not indicated animal is not more likely to be picked.more likely to be picked.

HH11: The independent : The independent variable variable doesdoes have an have an effect on the dependent effect on the dependent variable.variable. A grammatically A grammatically

indicated animal is more indicated animal is more likely to be picked.likely to be picked.

If HIf H00 is true, the kid has a 1/6th is true, the kid has a 1/6th chance (0.17) of getting one chance (0.17) of getting one right in each trial.right in each trial. So, given 5 tries, that’s a 40% So, given 5 tries, that’s a 40%

chance (.40) of getting one.chance (.40) of getting one. But odds of getting 3 are about But odds of getting 3 are about

3% (0.03), and odds of getting 4 3% (0.03), and odds of getting 4 are about .4% (0.0035).are about .4% (0.0035).

So, if the kid gets 3 of 5 right, So, if the kid gets 3 of 5 right, the likelihood that this came the likelihood that this came about by chance (Habout by chance (H00) are ) are slimslim..

=BINOMDIST(3, 5, 0.17, false)=BINOMDIST(3, 5, 0.17, false) Yields 0.03. 3 is number of Yields 0.03. 3 is number of

successes, 5 is number of tries, successes, 5 is number of tries, 0.17 is the probability of success 0.17 is the probability of success per try. True instead of false per try. True instead of false would be probability that would be probability that at most at most 3 were successes.3 were successes.

CriteriaCriteria In hypothesis testing, In hypothesis testing,

a criterion is set for a criterion is set for rejecting the null rejecting the null hypothesis.hypothesis.

This is a maximum This is a maximum probability that, if the probability that, if the null hypothesis were null hypothesis were true, we would have true, we would have gotten the observed gotten the observed result.result.

This has arbitrarily This has arbitrarily been (conventionally) been (conventionally) set to 0.05.set to 0.05.

So, if the probability So, if the probability pp of seeing what we of seeing what we see if Hsee if H00 were true were true is less than 0.05, we is less than 0.05, we reject the null reject the null hypothesis.hypothesis. If the kid gets 3 If the kid gets 3

animals right in 5 animals right in 5 trials, trials, pp=0.03 — that =0.03 — that is, is, p<p<0.05 so we 0.05 so we reject the null reject the null hypothesis.hypothesis.

Measuring thingsMeasuring things

When we go out into the world and When we go out into the world and measure something like reaction time for measure something like reaction time for reading a word, we’re trying to investigate reading a word, we’re trying to investigate the underlying phenomenon that gives rise the underlying phenomenon that gives rise to the reaction timeto the reaction time..

When we measure reaction time of reading When we measure reaction time of reading II vs. vs. theythey, we are trying to find out of there , we are trying to find out of there is a real, systematic difference between is a real, systematic difference between them (such that them (such that II is generally faster). is generally faster).

Measuring thingsMeasuring things

Does it take longer to read Does it take longer to read II than than theythey?? Suppose that in principle it takes Pat A ms to Suppose that in principle it takes Pat A ms to

read read II and B ms to read and B ms to read theythey.. Except sometimes his mind wanders, Except sometimes his mind wanders,

sometimes he’s sleepy, sometimes he’s hyper-sometimes he’s sleepy, sometimes he’s hyper-caffeinated.caffeinated.

Does it take longer for Does it take longer for peoplepeople to read to read II than than theythey??

Some people read/react slower than Pat. Some Some people read/react slower than Pat. Some people read/react faster than Pat.people read/react faster than Pat.

Normally…Normally…

Many things we measure, with their Many things we measure, with their noise taken into account, can be noise taken into account, can be described described (at least to a good approximation) by (at least to a good approximation) by this “bell-shaped” this “bell-shaped” normal normal distributiondistribution..

Often as we do statistics, we Often as we do statistics, we implicitly assume that this is the implicitly assume that this is the case…case…

Properties of the normal Properties of the normal distributiondistribution

A normal distribution can be described in A normal distribution can be described in terms of two parameters.terms of two parameters. = mean= mean = standard deviation (spread)= standard deviation (spread)

Interesting facts about Interesting facts about the standard deviationthe standard deviation

About 68% of the observations will be within About 68% of the observations will be within one standard deviation of the population one standard deviation of the population mean.mean.

About 95% of the observations will be within About 95% of the observations will be within two standard deviations of the population two standard deviations of the population mean.mean.

Percentile (mean 80, score 75, stdev 5): 15.9Percentile (mean 80, score 75, stdev 5): 15.9

Inferential statisticsInferential statistics

For much of what you’ll use statistics for, For much of what you’ll use statistics for, the presumption is that there the presumption is that there isis a a distribution out in the world, a truth of distribution out in the world, a truth of the matter.the matter.

If that distribution is a normal If that distribution is a normal distribution, there will be a population distribution, there will be a population mean (mean () and standard deviation () and standard deviation ().).

By measuring a sample of the population, By measuring a sample of the population, we can try to guess we can try to guess and and from the from the properties of our sample.properties of our sample.

A common goalA common goal Commonly what we’re after is an answer Commonly what we’re after is an answer

to the question: to the question: are these two things that are these two things that we’re measuring actually different?we’re measuring actually different?

So, we measure for So, we measure for II and for and for theythey. Of the . Of the measurements we’ve gotten, measurements we’ve gotten, II seems to be seems to be around A, around A, theythey seems to be around B, and seems to be around B, and B is a bit longer than A. The question is: B is a bit longer than A. The question is: given the inherent noise of measurement, given the inherent noise of measurement, how likely is it that we got that difference how likely is it that we got that difference just by chance?just by chance?

So, more or less, …So, more or less, …

If we knew the If we knew the actualactual mean of the mean of the variable we’re measuring and the variable we’re measuring and the standard deviation, we can be 95% standard deviation, we can be 95% sure that any given measurement we sure that any given measurement we do will land within two standard do will land within two standard deviations of that mean—and 68% deviations of that mean—and 68% sure that it will be within one.sure that it will be within one.

Of course, we can’t know the actual Of course, we can’t know the actual mean. But we’d like to.mean. But we’d like to.

EstimatingEstimating If we take a sample of the population and If we take a sample of the population and

compute the sample mean of the measures we compute the sample mean of the measures we get, that’s the best estimate we’ve got of the get, that’s the best estimate we’ve got of the population mean.population mean. =AVERAGE(A2:A10)=AVERAGE(A2:A10)

To estimate the spread of the population, we To estimate the spread of the population, we use a number related to the number of use a number related to the number of samples we took and the variance of our samples we took and the variance of our sample.sample. =STDEV(A2:A10)=STDEV(A2:A10) If you want to If you want to describe your sample describe your sample (that is if you (that is if you

have the entire population sampled), use STDEVP have the entire population sampled), use STDEVP instead.instead.

tt-tests-tests Take a sample from the population and measure Take a sample from the population and measure

it. Say you took it. Say you took nn measurements. measurements. Population estimates:Population estimates: = AVERAGE(sample), = AVERAGE(sample), = SQRT(VAR(sample)/ = SQRT(VAR(sample)/nn))

Your hypotheses determine what you expect your Your hypotheses determine what you expect your population mean to be if the null hypothesis is population mean to be if the null hypothesis is true.true. We’re actually considering variability in the We’re actually considering variability in the sample sample

meansmeans here—what is the mean mean you expect to here—what is the mean mean you expect to get, and what is the variance in those means?get, and what is the variance in those means?

You look at the distance of the sample mean You look at the distance of the sample mean from the estimated population mean (of sample from the estimated population mean (of sample means) and see if it’s far enough away to be very means) and see if it’s far enough away to be very unlikely (e.g., unlikely (e.g., pp<0.05) to have arisen by chance.<0.05) to have arisen by chance.

tt-tests-tests Does caffeine affect heart rate (example from Loftus Does caffeine affect heart rate (example from Loftus

& Loftus 1988)?& Loftus 1988)? Sample 9 people, measure their heart rate pre- and Sample 9 people, measure their heart rate pre- and

post-caffeination. The measure for each subject will post-caffeination. The measure for each subject will be the be the differencedifference score (post-pre). This is a within- score (post-pre). This is a within-subjects design.subjects design. Estimate the sample mean population: Estimate the sample mean population: MM=AVERAGE(B1:B10)=4.44=AVERAGE(B1:B10)=4.44MM=SQRT(VAR(B1:B10)/COUNT(B1:B10))=1.37=SQRT(VAR(B1:B10)/COUNT(B1:B10))=1.37

tt-score (like -score (like zz-score) is scaled (here, against estimated -score) is scaled (here, against estimated standard deviation), giving a measure of how “extreme” standard deviation), giving a measure of how “extreme” the sample mean was that we found.the sample mean was that we found.

If the If the tt-score (here 3.24) is higher than the criterion -score (here 3.24) is higher than the criterion tt (2.31, based on “degrees of freedom” = (2.31, based on “degrees of freedom” = nn-1 = 8) -1 = 8) and desired and desired -level (0.05), we can reject the null -level (0.05), we can reject the null hypothesis: caffeine affects heart rate.hypothesis: caffeine affects heart rate.

tt-tests: 2 sample means-tests: 2 sample means The more normal use of a The more normal use of a tt-test is to see if two -test is to see if two

sample means are different from one another.sample means are different from one another. HH00: : 11 = = 22

HH11: : 11 > > 22

This is a This is a directionaldirectional hypothesis—we are hypothesis—we are investigating not just that they are investigating not just that they are differentdifferent, , but that but that 11 is is moremore than than 22..

For such situations, our For such situations, our criterion tcriterion t score score should be should be one-tailedone-tailed. We’re only looking in . We’re only looking in one direction, and one direction, and 11 has to be has to be sufficiently sufficiently biggerbigger than than 22 to conclude that H to conclude that H00 is wrong. is wrong.

TailsTails If we are taking as our alternative hypothesis (HIf we are taking as our alternative hypothesis (H11) that ) that

two means simply two means simply differdiffer, then they could differ in either , then they could differ in either direction, and so we’d conclude that they differ if the direction, and so we’d conclude that they differ if the one were far out from the the other in either direction. If one were far out from the the other in either direction. If HH11 is that the mean will increase, then it is a directional is that the mean will increase, then it is a directional hypothesis, and then a one-tailed criterion is called for.hypothesis, and then a one-tailed criterion is called for.

tt-tests in Excel-tests in Excel

If you have one set of data in column If you have one set of data in column A, and another in column B,A, and another in column B,

=TTEST(A1:A10, B1:B10, 1, type)=TTEST(A1:A10, B1:B10, 1, type) Type is 1 if paired (each row in column Type is 1 if paired (each row in column

A corresponds to a row in column B), 2 A corresponds to a row in column B), 2 if independently sampled but with equal if independently sampled but with equal variance, 3 if independently sampled variance, 3 if independently sampled but with unequal variance.but with unequal variance.

Paired is generally better at keeping Paired is generally better at keeping variance under control.variance under control.

ANOVAANOVA

Analysis of Variance (ANOVA), finding Analysis of Variance (ANOVA), finding where the variance comes from.where the variance comes from.

Suppose we have three conditions and Suppose we have three conditions and we want to see if the means differ.we want to see if the means differ. We could do We could do tt-tests, condition 1 against -tests, condition 1 against

condition 2, condition 1 against condition 3, condition 2, condition 1 against condition 3, condition 2 against condition 3, but this condition 2 against condition 3, but this turns out to be not as good.turns out to be not as good.

Finding the varianceFinding the variance The idea of the ANOVA is to divide up the total The idea of the ANOVA is to divide up the total

variance in the data into parts (to “account for variance in the data into parts (to “account for the variance”):the variance”):

Within group variance (variance that arises Within group variance (variance that arises within a single condition)within a single condition)

Between group variance (variance that arises Between group variance (variance that arises between different conditions)between different conditions)

ANOVA:ANOVA: SSSS dfdf MSMS FF pp FcFcbetween groupsbetween groups …… 55 .... 2.452.45 0.0450.045

2.392.39within groupswithin groups …… 5454 ....totaltotal

Confidence intervalsConfidence intervals As well as trying to decide if your observed As well as trying to decide if your observed

sample is within what you’d expect your estimated sample is within what you’d expect your estimated distribution to provide, you can kind of run this distribution to provide, you can kind of run this logic in reverse as well, and come up with a logic in reverse as well, and come up with a confidence intervalconfidence interval::

Given where you see the measurements coming Given where you see the measurements coming up, they must be 68% likely to be within 1 CI of up, they must be 68% likely to be within 1 CI of the mean, and 95% likely to be within 2 CI of the the mean, and 95% likely to be within 2 CI of the mean, so the more measurements you have the mean, so the more measurements you have the better guess you can make.better guess you can make.

A 95% CI like 209.9 < µ < 523.4 means “we’re A 95% CI like 209.9 < µ < 523.4 means “we’re 95% confident that the 95% confident that the realreal population mean is in population mean is in there”.there”. =CONFIDENCE(0.05,=CONFIDENCE(0.05,

STDEV(sample),STDEV(sample),COUNT(sample))COUNT(sample))

Correlation and Chi Correlation and Chi squaresquare

Correlation between Correlation between two two measured two two measured variables is often variables is often measured in terms of measured in terms of (Pearson’s) r.(Pearson’s) r.

If r is close to 1 or -1, If r is close to 1 or -1, the value of one the value of one variable can predict variable can predict quite accurate the quite accurate the value of the other.value of the other.

If r is close to 0, If r is close to 0, predictive power is low.predictive power is low.

Chi-square test is Chi-square test is supposed to help supposed to help us decide if two us decide if two conditions/factors conditions/factors are independent of are independent of one another or not. one another or not. (Does knowing one (Does knowing one help predict the help predict the effect of the effect of the other?)other?)

So…So…

There’s still work to be done. Since There’s still work to be done. Since I’m not sure exactly I’m not sure exactly whatwhat work that work that is, once again… no lab work to do.is, once again… no lab work to do.

Places to go:Places to go: http://http://davidmlanedavidmlane.com/.com/hyperstathyperstat// http://www.stat.sc.edu/webstat/http://www.stat.sc.edu/webstat/