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Lecture 13: Statistics

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MA/MSc LVC

U. of York

Autumn 2007

Bill Haddican

Greg Guy

Outline:

Conventions in presenting quantitative data

Tests of significance-when to use which?

Chi-square tests

Goldvarb

T-tests (seminar)

Pearson’s r (seminar)

Greg Guy

1. Tables must always have a title (legend). Ideally, the title should be descriptive enough that your reader doesn’t have to read the text to interpret it. That is, your table should be able to stand alone.

- Tables should also be numbered. The number precedes the legend:
Table 1. Frequencies of use of be like by speaker sex.

3. Figures in columns should lined up by decimal places.

4. Provide N’s as well as %’s when possible in your tables.

(Source: my diss.)

5.You should refer to all tables in your text.

Additional information needed to interpret the table--such as notes on differences in significance go below the table in a footnote.

Figures must include units of measurement.

On figures, the independent variable goes on the x axis and the dependent variable goes on the y-axis.

If your independent variable is a category variable use a bar-graph.

From Tagliamonte and Hudson 1999:162

Figure X: Coda r deletion in NYC by style

From Labov 1966

10.Cardinal rule for tables: Frequencies should reflect use of the dependent variable as a proportion of the total number of tokens of the independent variable (not the other way around.)

Table X: Use of be like vs. say by speaker sex. (BAD Table!)

- The %s here tell us that 33% of our be like tokens are by men and 67% are by women. Our say tokens are evenly distributed by sex.
- But this isn’t what we want to know. Rather, what we want to know is whether men use be like vs. say to a greater or lesser extent than women. To see this, we need to look at use of be like vs. say among men as a proportion of the total number of tokens for men.

Table X: Use of be like vs. say by speaker sex. (GOOD Table!)

- The %’s here tell a very different story. Here, we see that Women tend toward be like much more strongly than men.

- Quantitative sociolinguistic work involves positing relationships between variables.

(Source: my diss.)

- But how confident can we be that such distributions really do reflect a relationship between our variables and are not just by chance.
- Tests of significance, then, are used to provide an estimate of this chance.

- Tests of significance weigh two competing hypotheses:
- The null hypothesis: There is NO relationship between the dependent variable and the independent variable.
- The experimental hypothesis: There IS a relationship between the dependent variable and the independent variable.

- Example: speaker sex and be like usage.
- The null hypothesis: There is NO relationship between speaker sex and be like usage.
- The experimental hypothesis: There IS a relationship between speaker sex and be like usage.

- Our test of significance--a chi-square test in this case--will help us decide if observed differences in be like use by speaker sex reflect a relationship or are coincidental.
- Tests of significance generate a probability value, denoted as p. This is the probability that the null hypothesis is correct.
- Our p-value is the chance that there is NO relationship between our variable.

- In other words the smaller our p, the greater the chance that there IS a relationship between our variables.
- p=.05, for example, indicates a 5%, or 1/20 chance that the relationship between our variables is a fluke.
- p=.01 indicates a 1% or 1/100 chance that the observed relationship is accidental.
- p=.001 indicates a .1% or 1/1000 chance that the observed relationship is accidental.

- (These are some standard benchmarks used.)

- In sociolinguistics and in other social sciences p=.05 is a standard minimum threshold for positing a relationship.
- In multivariate analyses that you’ve seen, for example, when a given figure is said to be “not significant” this means p>.05.

Table 4.: Significant factor groups favoring (non-standard) participial affix doubling

Factor GroupFrequencyWeight

Educational attainment

High 141/29248%.25Medium 187/20889%.81

Low 186/21587%.52

Sex

Women229/29677%[.61]

Men285/41968%[.42]

(Source: my diss.)

- In your reading of sociolinguistic work, you’ll have noticed different kinds of tests of significance: chi-square tests, t-tests, F-tests. There are others, but these are some of the most frequently used.
- Which of these to use depends on the kind of data you have.
- Ordinal variables are categorical, e.g. male vs. female, Labour vs. Tory vs. Liberal.
- Continuous variables are numerical or quantitative, e.g. formant frequencies, temperatures, heights.

- Chi-square tests are used for two nominal variables.

Use of be like vs. say by speaker sex

- t-tests are used with a nominal variable and a continuous variable. More precisely, they compare the means of two samples.

F1s by social class

- A good question that’s undoubtedly in everyone’s head right now: What kind of test should be use with two quantitative variables. Say, for example, that we want to look at F1 by speaker age?
- Typically, measurements of correlation are used in such cases. These indicate to what degree one variable predicts or covaries with another.

How it works.

- What a chi-square test does is test the null hypothesis, that is, that there is NO relationship between our variables.
- It compares observed values in a distribution with the expected values and measures the probability that the difference in these two is by chance.

How it works.

- The observed values are what we have in our data, which, let’s suppose, is the following.
- Observed values: Use of be like vs. say by speaker sex

How it works.

- How, then do we determine the expected values? First, look at the totals for be like and say. How would we expect them to be distributed if there were no relationship?
- Expected values:Use of be like vs. say by speaker sex

How it works.

- How, then do we determine the expected values? First, look at the totals for be like and say. How would we expect them to be distributed if there were no relationship?
- Expected values:Use of be like vs. say by speaker sex

How it works.

- Now, in the previous example, figuring out the expected values was easy because the number of tokens for men and women was the same. What would we do if it wasn’t?

How it works.

- Now, in the previous example, figuring out the expected values was easy because the number of tokens for men and women was the same. What would we do if it wasn’t?
- Observed values:Use of be like vs. say by speaker sex

How it works.

- Easy. The expected values for each cell will be:
((∑column)(∑row))/total.

Expected values:Use of be like vs. say by speaker sex

How it works.

- And so on.
Expected values:Use of be like vs. say by speaker sex

How it works.

- Once you calculate one cell’s value in this way, you can calculate the rest by subtracting from the marginals.
Expected values:Use of be like vs. say by speaker sex

How it works.

- Once filled out, our table of expected values will look like this.
Expected values:Use of be like vs. say by speaker sex

How it works.

- We then compare the observed and the expected values in each cell. Note that the difference in each case is 6.15 (absolute value).

How it works.

- To figure out what chance there is that this difference is by chance we use the following formula.
- ∑(observed-expected)2/expected
- This means three steps:
- We square the difference between the observed and expected values in each cell
- We divide this number by the expected value for each cell
- We then add all of these cell values together.

How it works.

- Let’s do this step by step.
- First, subtract the expected from the observed for each cell.

How it works.

- Second, square these differences.

How it works.

- Third, divide these squares by the expected values for each cell.

How it works.

- Finally, add all of these values together and this is our chi-square value.
- X2=.46+.74+.57+.93=2.70

How it works.

- We then look up our chi-square on a chi-square table, which will give the probabilities (p-values) associated with each chi-square. (Such a table can be found on the web or in the back of any statistics book.)
- To do this we will need to know the degrees of freedom (d.f.) The degrees of freedom for a chi-square is (#rows-1)(#columns-1. Recall that we have a 2 x 2 table, so our d.f.= (2-1)(2-1)=1.

How it works.

- Looking in our table, we find the following at 1 d.f.:
chi-squarep

3.84.05

5.41.02

6.64.01

10.33.001

How it works.

- Our chi-square value of 2.70, then, means that there’s more than a 5% chance that our observed relationship is by chance! (In fact, it’s about 10%.)
- At the level of p=.05, then, we do not reject the null hypothesis.
- NB: chi-square tests cannot be performed when the expected frequency of any given cell is less than 5 and not ideal when total N < 20. Instead use Fisher’s exact test.

Other ways to do chi-square tests

Excel

- On a spread sheet, you’ll need a table of observed values and a table of expected values, as above.
- Click on a free cell. This is where your result will appear.
- Then, on the “Insert” menu at the top, select “Function.” A dialog box will then appear with two columns in it.
- In the left column select “statistical.” In the right column, select “CHITEST.”

Other ways to do chi-square tests

Excel

- A new dialog box will appear with two fields, one asking you for a range of observed values and one asking for a range of expected values.
- You can input these by right-clicking and dragging the cursor over the relevant tables in your spreadsheet, and selecting “OK.”
- Excel will then give you the p-value.

Other ways to do chi-square tests

Web pages

- An even easier solution is to use webpages such as the following:
- http://www.graphpad.com/quickcalcs/contingency1.cfm
- No explanation necessary for this!

- Goldvarb (Varbrul) is a kind of multivariate analysis (specifically, a logistic regression model).
- In the kind of variation data that we typically work with, different kinds of factors combine to produce the patterns of variation we see.
- A speaker’s use of t-glottaling, say, may be influenced by his/her age, gender, dialect, as well as linguistic factors such as preceding and following segment.
- The problem, then, is to sort out the effect of these competing constraints on variation.

- What Goldvarb does is build a model of this variation that estimates the contribution of different factors on the dependent variable.
- Variables correspond (roughly) to factor groups, and the different categories of these variables are called factors. The factor group sex, for example, will (presumably) have the factors male and female.
- Each factor in each group is then assigned a weight which estimates its contribution to the application value-a variant of the dependent variable.

Source: my diss.

- In reporting goldvarb analyses, authors typically also provide the N’s and the input (also corrected mean or overall tendency). This is roughly the overall likelihood of occurrence of the application value.
- It is also typical to report N’s and frequencies for each factor.
- Note, also, that goldvarb is only used with non-categorical variables. Variables for which variation is categorical or near categorical >~95% are excluded.

Goals:

- To review some conventions of presenting quanitative data.
- To show when to use different kinds of tests of significance.
- To show how to perform a chi-square test.

Guy, Gregory. 1993. The quantitative analysis of linguistic variation. (photocopy pack.)

Labov, William. 1966. The social stratification of English in New York City. Washington D.C.: Center for Applied Linguistics.

Tagliamonte, Sali and Rachel Hudson. 1999. Be like et al beyond America. The quotative system in British and Canadian English. Journal of Sociolinguistics 3:147-172.

Garson webpage (nice explanation of Fisher’s exact test):

http://www2.chass.ncsu.edu/garson/PA765/fisher.htm