CE3502. Environmental Measurements, Monitoring & Data Analysis. ANOVA: Analysis of. T-tests: Excel options

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1 CE350. Environmental Measurements, Monitoring & Data Analysis ANOVA: Analysis of Variance T-tests: Excel options Paired t-tests tests (use s diff, ν =n=n x y ); Unpaired, variance equal (use s pool, ν = n x +n y ) Unpaired, variance not equal σ σ d i = x y variance V x y = + n n degrees of freedom = ν = x y s s x y + nx n y ν = s s x y n n x y + n 1 n 1 x y 1

2 Excel output t Test: Two Sample Assuming Unequal Variances Variable 1 Variable Mean Variance Observations 9 10 Hypothesized Mean Difference 0 df 9 t Stat P(T<=t) one tail t Cii Critical one tail P(T<=t) two tail t Critical two tail.6157 Motivation: ANOVA Comparison of sets of measurements, or one set of measurements and a fixed value (e.g., regulatory limit) can be done with t-tests. In many cases, it is desirable to compare multiple sets of data to assess whether they are all part of the same population or whether hth they differ (ttiti (statistically) ll)from one another. For such comparisons, the most common technique is ANOVA.

3 Examples: ANOVA 1. Multiple groundwater wells have been sampled to determine whether any are contaminated (several replicates at each well);. Samples are collected along multiple transects in Lake Superior to determine if horizontal spatial differences exist; 3. Air samples are collected on multiple days under different weather conditions to determine if air quality varies systematically with weather conditions; 4. Multiple soil samples are collected within several areas of a Brownfield site to determine if any areas are contaminated. ANOVA Null Hypothesis: There are no significant differences among the groups (i.e., they are all part of the same population) 3

4 Theory: ANOVA Analysis of Variance (ANOVA) consists of the following steps: 1. Test whether groups of data meet assumptions required to perform ANOVA (lack of heteroscedasticity);. Assess variability ( noise ) within individual groups of data; 3. Assess variability between groups of data; 4. Compare within-group and between-group variability; 5. If within-group and between-group variability is of a comparable magnitude, the groups are deemed to be not different or to belong to the same population. Preliminary test: Heteroscedasticity Parametric ANOVA assumes that groups of data exhibit similar within-group variance. Heteroscedasticity is the property of exhibiting different variances. Tests of heteroscedasticity include: 1. Box plots. Levene s test 3. Bartlett s test 4. Probability plot of residuals If these tests fail to meet acceptable criteria, either (1) the data must be transformed (e.g., log-transformation) and retested, or () a non-parametric ANOVA must be performed. 4

5 ANOVA. Step 1. Test for Heteroscedasticity Calculate means for each group, and residuals (x ij -μ i ) Homoscedasticity Probability Plot 5

6 ANOVA: Step Consider a set of measurements including k groups each with n j measurements (x i,j ): The variance for each group is defined as usual: s j = n j ( x ) i, j x j n i = 1 j 1 Group average From these variances, a pooled within-group variance k may be obtained: ( nj 1) sj The degrees of freedom is: j= 1 sw = k ν = n 1 ( n 1) j j= 1 j Anova: Step The between-group variance is calculated as: s b = k j= 1 j ( ) j n y y k 1 Average of all samples from all groups ν = k 1 Degrees of freedom Degrees of freedom If the groups do not differ from one another, s b should be ~ equal to s w. 6

7 ANOVA: Step 3 Comparison of variances is performed with F-test: F s = s b w F crit value is read from Table A.7 as F df1, df, α If F is less than F crit then variances are equal and groups are not statistically different from one another. 7

8 Trouble spots If n 1 or n are not in F table, either go to Web and use F crit calculator or interpolate; How does one remember which is on top, s b or s w? F ratio is always > 1; which is bigger, s b or s w? y gg b w b comes before w and goes on top; 8

9 ANOVA in Practice: The procedure just outlined is correct, but it is not the one most often followed. The procedure outlined in Navidi (006) is more amenable to automation and is the one most commonly employed. Within Excel (as well as within many other readily-available software packages), an ANOVA routine is available. It is well worth your while to learn this procedure and how to interpret the output. The procedure is available under the pulldown menu Tools, Data analysis, Anova: Single factor Example 1: ANOVA Fine aerosols were measured in 3 rooms with the following results (conc. in µg/m 3 ). Room A Room B Room C Assess whether any room differs significantly from the others. 9

10 Example 1: cont d In Excel we choose Anova: Single Factor, highlight the area with the input data, select an area for the output, and click on OK. The output looks like: Anova: Single Factor SUMMARY Groups Count Sum Average Variance Room A Room B Room C ANOVA Source of Variation SS df MS F P-value F crit Between Groups Within Groups Total Anova: Single Factor ANOVA output SUMMARY Groups Count Sum Average Variance Room A Room B Room C ANOVA Source of Variation SS df MS F P-value F crit Between Groups Within Groups Total Because F > F crit, we can conclude with greater than 99% certainty that these 3 rooms do NOT all have similar aerosol concentrations. 10

11 Beyond ANOVA ANOVA tells us only if the groups are similar. If they are not all similar, ANOVA does not identify which of the groups is dissimilar from the others. How can we determine which groups are similar and which are dissimilar? t-tests (available in Excel) Bonferroni t-tests (available in statistics software) Example 1: Which are different? t-test: Two-Sample Assuming Equal V Room A Room B Mean Variance Observations 4 4 Pooled Variance Hypothesized Mea 0 df 6 t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail Room A Room C

12 Example 1: concluded Room B Room C Room C Room B Mean Variance Observations 4 4 Pooled Variance Hypothesized Mea 0 df 6 t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail Beyond 1-way ANOVA If heteroscedasticity tests fail, either transform data or use non-parametric test; If there are interactions between groups (e.g., season and % sunny days), use -way ANOVA; To identify which groups are similar/dissimilar, use t-tests, Bonferroni s t- tests, or a host of other available tests. 1

13 Example. Indoor CO CO Conc. (ppm) Dow Atrium 00 Dow 85 Dow 8th floor office 0 15:30:14 15:30:58 15:31:41 15:3:4 15:33:07 15:33:50 15:34:34 15:35:17 15:36:00 Time Anova: Single Factor SUMMARY Groups Count Sum Average Variance Atrium Office ANOVA Source of Variation SS df MS F P-value F crit Between Groups E-68.6 Within Groups Total