1. CENTRE FOR INNOVATION, RESEARCH AND COMPETENCE IN THE LEARNING ECONOMY Session 2: Basic techniques for innovation data analysis. Part I: Statistical inferences and comparisons of groups Taehyun Jung taehyun.jung@circle.lu.se CIRCLE, Lund University 10:30-12:00 December 10 2012 For Survey of Quantitative Research, NORSI

2. CIRCLE, Lund University, Sweden 2 Objectives of this session

3. CIRCLE, Lund University, Sweden  Correlation  Statistical Inference and Hypothesis Testing  t-Test  Confidence Interval  Chi-square Statistic 3 Contents

4. CIRCLE, Lund University, Sweden Correlation 4

5. CIRCLE, Lund University, Sweden  A scatterplot displays the strength, direction, and form of the relationship between two quantitative variables. – As in any graph of data, look for the overall pattern and for striking departures from that pattern.  Form: linear, curved, clusters, no pattern  Direction: positive, negative, no direction  Strength: how closely the points fit the “form” – An important kind of departure is an outlier, an individual value that falls outside the overall pattern of the relationship. Scatterplot 5

6. CIRCLE, Lund University, Sweden 6 Linear Nonlinear No relationship

7. CIRCLE, Lund University, Sweden  With a strong relationship, you can get a pretty good estimate of y if you know x.  With a weak relationship, for any x you might get a wide range of y values. The strength of the relationship between the two variables can be seen by how much variation, or scatter, there is around the main form. 7

8. CIRCLE, Lund University, Sweden  The correlation coefficient r measures the strength of the linear relationship between two quantitative variables. – r is always a number between -1 and 1. – r > 0 indicates a positive association. – r < 0 indicates a negative association. – Values of r near 0 indicate a very weak linear relationship. – The strength of the linear relationship increases as r moves away from 0 toward -1 or 1. – The extreme values r = -1 and r = 1 occur only in the case of a perfect linear relationship. – Part of the calculation involves finding z, the standardized score  Allows us to compare correlations between data sets where variables are measured in different units or when variables are different The sample Pearson’s correlation coefficient r measures the strength of the linear relationship between two quantitative variables. 8 correlation

9. CIRCLE, Lund University, Sweden  Correlation makes no distinction between explanatory and response variables.  r has no units and does not change when we change the units of measurement of x, y, or both.  Positive r indicates positive association between the variables, and negative r indicates negative association.  The correlation r is always a number between -1 and 1.  Cautions – Correlation requires that both variables be quantitative. – Correlation does not describe curved relationships between variables, no matter how strong the relationship is. – Correlation is not resistant. r is strongly affected by a few outlying observations. – Correlation is not a complete summary of two-variable data. Facts About Correlation 9

10. CIRCLE, Lund University, Sweden  Strength: How closely the points follow a straight line.  Direction is positive when individuals with higher x values tend to have higher values of y “r” ranges from − 1 to +1 10

11. CIRCLE, Lund University, Sweden  Correlations are calculated using means and standard deviations and thus are NOT resistant to outliers.  Just moving one point away from the general trend here decreases the correlation from −0.91 to −0.75. Influential points 11

12. CIRCLE, Lund University, Sweden Statistical Inference and Hypothesis Testing 12

13. CIRCLE, Lund University, Sweden  The normal distribution has the bell-shaped (Gaussian) form. – arises from the central limit theorem, which states that under mild conditions, the mean of a large number of random variables independently drawn from the same distribution is distributed approximately normally, irrespective of the form of the original distribution – very tractable analytically Normal distribution 13

14. CIRCLE, Lund University, Sweden Testing a hypothesis relating to the population mean 14

15. CIRCLE, Lund University, Sweden 15

16. CIRCLE, Lund University, Sweden  four standard deviations above the hypothetical mean? – the chance of getting such an extreme estimate is only 0.006%. – We would reject the null hypothesis  The usual procedure for making decisions is to reject the null hypothesis if it implies that the probability of getting such an extreme sample mean is less than some (small) probability p. – For example, the probability of getting such an extreme sample mean is less than 0.05 (5%) – The 2.5% tails of a normal distribution always begin 1.96 standard deviations from its mean 16

17. CIRCLE, Lund University, Sweden 17 Type I error: rejection of H 0 when it is in fact true. Probability of Type I error: in this case, 5% Significance level (size) of the test is 5%.

18. CIRCLE, Lund University, Sweden We can of course reduce the risk of making a Type I error by reducing the size of the rejection region. 18

19. CIRCLE, Lund University, Sweden t-Test 19

20. CIRCLE, Lund University, Sweden What if we do not know the standard deviation? The test statistic has a t distribution instead of a normal distribution 20 s.d. of X known discrepancy between hypothetical value and sample estimate, in terms of s.d.: 5% significance test: reject H 0 :  =  0 if z > 1.96 or z < –1.96 s.d. of X not known discrepancy between hypothetical value and sample estimate, in terms of standard error (s.e.): 5% significance test: reject H 0 :  =  0 if t > t crit or t < –t crit

21. CIRCLE, Lund University, Sweden  For a sample of size n, the sample standard deviation s is: – n − 1 is the “degrees of freedom.” – The value s/√n is called the standard error of the mean SEM. – Scientists often present their sample results as the mean ± SEM. 21

22. CIRCLE, Lund University, Sweden  When the number of degrees of freedom is large, the t distribution looks very much like a normal distribution (and as the number increases, it converges on one)  Then, why t-dist? – Although the distributions are generally quite similar, the t distribution has longer tails than the normal distribution, the difference being the greater, the smaller the number of degrees of freedom – the rejection regions have to start more standard deviations away from zero for a t distribution than for a normal distribution 22

23. CIRCLE, Lund University, Sweden Example 23

24. CIRCLE, Lund University, Sweden  The t tests are exactly correct when the population is distributed exactly normally. However, most real data are not exactly normal.  The t tests are robust to small deviations from normality. This means that the results will not be affected too much. Factors that do strongly matter are: – Random sampling. The sample must be an SRS from the population. – Outliers and skewness. They strongly influence the mean and therefore the t procedures. However, their impact diminishes as the sample size gets larger because of the Central Limit Theorem. – Specifically:  When n < 15, the data must be close to normal and without outliers.  When 15 > n > 40, mild skewness is acceptable, but not outliers.  When n > 40, the t statistic will be valid even with strong skewness. Robustness 24

25. CIRCLE, Lund University, Sweden Confidence interval 25

26. CIRCLE, Lund University, Sweden  Any hypothesis lying in the interval from   min to   max would be compatible with the sample estimate (not be rejected by it). We call this interval the 95% confidence interval. Confidence interval 26 (1)(2)  min  min +sd X  min –sd  min –1.96sd  max  max +1.96sd  max –sd  max +sd

27. CIRCLE, Lund University, Sweden 27

28. CIRCLE, Lund University, Sweden Chi-square statistic 28

29. CIRCLE, Lund University, Sweden  STATA command –. tab dused_ndef largef, col chi – Pearson chi2(1) = 11.4978 Pr = 0.001 Can we conclude that large firms use patents more strategically than small firms based on this table? 29 Use of patents by firm size Small FirmLarge FirmColumn total Non-strategic use #1601,1131,273 % column91.9581.6682.82 Strategic use #14250264 % column8.0518.3417.18 Total#1741,3631,537 % column100.00

30. CIRCLE, Lund University, Sweden Chi-square hypothesis test 30

31. CIRCLE, Lund University, Sweden  We want to test the hypothesis that there is no relationship between these two categorical variables (H 0 ). – To test this hypothesis, we compare actual counts from the sample data with expected counts given the null hypothesis of no relationship. – The expected count in any cell of a two-way table when H 0 is true is:  The chi-square statistic (  2 ) is a measure of how much the observed cell counts in a two-way table diverge from the expected cell counts. 31

32. CIRCLE, Lund University, Sweden 32

33. CIRCLE, Lund University, Sweden  For the chi-square test, H 0 states that there is no association between the row and column variables in a two-way table. The alternative is that these variables are related.  If H 0 is true, the chi-square test has approximately a χ2 distribution with (r − 1)(c − 1) degrees of freedom. 33 The P-value for the chi-square test is the area to the right of  2 : P(χ 2 ≥ X 2 ).

34. CIRCLE, Lund University, Sweden  Probability of rejecting the null hypothesis if H0 is true – Typically,.05 or.01 significance level – With a significance level of.05 and 1 df, X2=3.84; we will reject H0 when X2* is greater than 3.84 and accept H0 when X2* is less than 3.84. – if the null hypothesis is true (if the variables are not related in the population), we will still (incorrectly) reject H0 (conclude that the variables are related in the population) about 5 times (or 1 time) in 100 hypothesis tests  A key step in the hypothesis test is deciding how willing we are to make a Type I error. (We must take some chance of rejecting a true null hypothesis or we will have no chance of rejecting a false one.) – Type I error: incorrectly rejecting the null hypothesis. – Type II error: Incorrectly accepting the null hypothesis. Significance Level (alpha) 34