R in Linguistic Analysis. Wassink 2012 University of Washington Week 6

Transcription

1 R in Linguistic Analysis Wassink 2012 University of Washington Week 6

2 Overview R for phoneticians and lab phonologists Johnson 3 Reading Qs Equivalence of means (t-tests) Multiple Regression Principal Components Analysis I Explore package: Vowels.R

3 Reading questions [JR] In section 3.1.4,...What statistical options are open to us when we have data that is mixed, i.e. some of it is paired, and some of it isn't? I'm guessing we'd have to settle for the standard t-test (or some other test entirely), since <100% of the data is actually paired... [MO] I'm confused about what Johnson is showing in the box on pages He says in the box that the variable "year" is being treated as a continuous variable when he wants to be nominal... But it looks like the summary() output is the same on both pages. Am I overlooking something obvious about what has changed? [NK] (notes box) Johnson says that some coefficients might not be significantly different from zero, but would still be a reliable predictor when different models are compared with each other. If this is the case, what is the value of performing the t-test at all? Is there a threshold for the t-test below which the coefficient would not be selected even if it was a reliable predictor for comparing different models? [ABW] The examples for running t-tests require 2 levels for our factors. Sometimes, we wish to choose two levels of a variable with >2 levels. Can we use the col.vars argument to do this? [ABW] How to Create a table of publishable quality from R table?

4 Norm norm1.php sample dataset: 535testPNWEdata.txt 535testPNWEdata.txt

5 Typical Workflow (1) What is my research question? (2) How do I read my data into R? What is the structure of my dataset? - can I assume normality? independence of observations? homogeneity of variance? How do I look at just the first few lines to determine this? (3) Install the packages I need in R (4) How should I operationalize my variables? Which is/are the dependent variable(s), and which is/are the independent variable(s)? What type of variables are they? (Categorical, Ord, Int., Ratio) - How do I ensure R treats them as the appropriate type? (5) What is the best way to represent my data graphically? What plots do I need to understand the structure of the data? (6) What are my hypotheses regarding the phenomenon I'm interested in? How do I test the hypotheses I have? What do I want to know? typical values? Means similarity of means? t-tests (7) What graphics do I need to export? How to create a formant table

6 Our Workflow

7 Goals of Quantitative Analysis [J1] 1. Data reduction: summarize trends (mean, dispersion), capture patterns among variables. mean() sd() cor.test() 2. Inference: generalize from a representative set of observations (sample) to a larger universe of possible observations t.test() aov() 3. Discovery of relationships between variables: find descriptive or causal patterns in data (mult. regression, factor analysis) 4. Exploration of processes that may have a basis in probability: theoretical modeling, sentence parsing We have to balance huge studies that address tiny questions against small studies that cover a wider range of interesting issues.

8 Structure of the dataset Determine factors present in dataframe Coerce variables into appropriate type Attach() Factor()

9 Hypotheses H0: Means for F1(ae) and F1(ey) are equivalent. H1: Means for F1(ae) and F1(ey) are not equivalent. our alternative hypothesis is non-directional: agnostic regarding whether meanf1(ae) is higher or lower than meanf1(ey). = non-directional.

10 Normality Kolmogorov-Smirnov and Shapiro-Wilk: specifically test whether sample scores come from a normal distribution (p<.05 here indicates a *nonnormal* distribution) shapiro.test(x50_f1[vowel_code=="ae"]) Shapiro-Wilk normality test data: X50_F1[vowel_code == "ae"] W = 0.979, p-value = 1.371e-07

11 Homogeneity of variance as you explore the structure of all levels of your variable, the variances are the same. look at variances and check, or use Fligner-Kileen, Ansari-Bradley, for non-normally-distributed data, or Levene s test for normally-distributed data (interpret p as for K-S test)

12 t-test When RQ requires testing equivalence of means students t-test of independent samples=parametric test Mann-Whitney U-test = nonparametric (unpaired) Wilcoxon test = nonparametric (paired) When you have prior knowledge about direction of possible difference, the result you need for a significant finding can be less extreme than for alternative hypothesis whose direction you cannot predict. one-tailed tests, less extreme p-value

13 t-test D.V. is a Ratio-scaled variable (F1) I.V. is a Categorical variable (vowel_code) Mann-Whitney U-test (for unpaired data): > test<-wilcox.test(x50_f1[vowel_code=="ae"],x50_f1[vowel_code=="ey"], paired=f, correct=t, alternative="greater"); options(scipen=10); print(test) Wilcoxon rank sum test with continuity correction data: X50_F1[vowel_code == "ae"] and X50_F1[vowel_code == "ey"] W = , p-value < 2.2e-16 alternative hypothesis: true location shift is greater than 0

14 Coefficient of Correlation association between continuous ratio or interval-scaled data -1 to +1 (0=no association) sign relates to the direction of a correlation the value (absolute size) relates to the size or strength of the correlation

15 Johnson (2008)

16 Covariance you may recall: variance is squared, unsigned average distance from the mean covariance: a measure of the average relationship between two variables. Measured as the average cross-product deviation (the cross-product / n-1) cov(x,y)=!!!!!!!!!!!!!!!!!

17 Correlation Coefficient Covariance heavily dependent on scale We can convert the covariance into standard units by standardization (to the standard deviation) r=!!!!!!!!!!!!!!!!!!!!!!!! Correlation is identical to covariance, except that correlation is scaled by standard deviations.

18 multiple regression When RQ leads us to investigate the possibility of a linear relationship between two (or more) numerical variables e.g., Are the F1 and F2 values correlated for speaker SB1? subset() to constrain data to single speaker Pearson s product moment correlation, r We may examine correlations between variables one-by-one, or we may design a model using combinations of variables, and investigate which combination of variables best predicts our outcome variable.

19 Log-likelihood AIC (Akaike Information Criterion) approach get as good a fit as possible with a minimum # of predictor variables maximize the likelihood of the model penalize addition of each new parameter added to the model > summary(pnwe.reg.test<-step(lm(x50_f1~1,,data=sb1),x50_f1~x50_f2)) Start: AIC= X50_F1 ~ 1 Df Sum of Sq RSS AIC + X50_F <none> Step: AIC= X50_F1 ~ X50_F2 Df Sum of Sq RSS AIC <none> X50_F Call: lm(formula = X50_F1 ~ X50_F2, data = SB1) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) < 2e-16 *** X50_F *** --- Signif. codes: 0 *** ** 0.01 * Residual standard error: on 805 degrees of freedom (5 observations deleted due to missingness) Multiple R-squared: , Adjusted R-squared: F-statistic: on 1 and 805 DF, p-value:

20 PCA Also used for data reduction (determine underlying patterns) Principal Components Analysis may be thought of as a series of multiple regression analyses We use data drawn from the covariance matrix Where we see correlations between variables, we use these to define abstract components that help account for variation in the data. Sometimes we have more variables than we need (variables tell the same story ), and can use one (or combine the two into 1 factor)

21 PCA loadings matrix: our set of principal components (Z-scores) scores matrix: the outcome of each data value We can reconstruct our original data matrix from these two matrices > summary(pc<-princomp(~ X50_F1, X50_F2, data=sb1)) Importance of components: Comp.1 Standard deviation Proportion of Variance Cumulative Proportion 1.000