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1 CHAPTER 5: PARAMETER ESTIMATION 5-0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be
2 CHAPTER 5: PARAMETER ESTIMATION 5-1 CHAPTER 5: PARAMETER ESTIMATION 1 Estimation Methods 1.1 Introduction 1.2 Estimation by the Method of Moments 1.3 Estimation by the Method of Maximum Likelihood 2 Properties of Estimators 2.1 Introduction 2.2 Unbiased 2.3 Efficiency 2.4 Consistency
3 CHAPTER 5: PARAMETER ESTIMATION Confidence Intervals 3.1 Introduction understanding the concept 3.2 Finding confidence intervals in practice Pivotal method 3.3 One-sample problems Confidence Intervals for Confidence Intervals for 3.4 Two-sample problems Confidence Interval for Confidence Interval for 3.5 Summary
4 CHAPTER 5: PARAMETER ESTIMATION Estimation Methods 1.1 Introduction Aims of biological research (Quinn and Keough 2002)
5 CHAPTER 5: PARAMETER ESTIMATION 5-4 Relationships measures of association (Quinn and Keough 2002)
6 CHAPTER 5: PARAMETER ESTIMATION 5-5 Cause versus effect: an example There is a correlation between the number of roads built in Europe and the number of children born in the United States. o Does that mean that if we want fewer children in the U.S., we should stop building so many roads in Europe? o Or, does it mean that if we don't have enough roads in Europe, we should encourage U.S. citizens to have more babies? o Of course not. (At least, I hope not). While there is a relationship between the number of roads built and the number of babies, we don't believe that the relationship is a causal one.
7 CHAPTER 5: PARAMETER ESTIMATION 5-6 Cause versus effect This leads to consideration of what is often termed the third variable problem. In the example above, it may be that there is a third variable that is causing both the building of roads and the birthrate that is causing the correlation we observe. [such a variable is called a confounding variable or a confounder] For instance, perhaps the general world economy is responsible for both. When the economy is good more roads are built in Europe and more children are born in the U.S. The key lesson here is that you have to be careful when you interpret correlations
8 CHAPTER 5: PARAMETER ESTIMATION 5-7 Cause versus effect Likewise, if you observe a correlation between the number of hours students use the computer to study and their grade point averages (with high computer users getting higher grades), you cannot assume that the relationship is causal: that computer use improves grades. In this case, the third variable might be socioeconomic status -- richer students who have greater resources at their disposal tend to both use computers and do better in their grades. It's the resources that drives both use and grades, not computer use that causes the change in the grade point average. This complicates making valid inferences!!!
9 CHAPTER 5: PARAMETER ESTIMATION 5-8 Types of relationships No relationship, positive relationship, negative relationship (see before) These are the simplest types of relationships we might typically estimate in research. The pattern of a relationship can be more complex than this, and one can try to find the most appropriate model to capture the observed pattern.
10 CHAPTER 5: PARAMETER ESTIMATION 5-9 Natural experiments Natural experiments cannot completely rule out alternative explanations of the observed associations: confounding cannot entirely be ruled out (Quinn and Keough 2002) Do you know the difference between correlation and association?
11 CHAPTER 5: PARAMETER ESTIMATION 5-10 Manipulative experiments (Quinn and Keough 2002)
12 CHAPTER 5: PARAMETER ESTIMATION 5-11 Statistical inference (Quinn and Keough 2002)
13 CHAPTER 5: PARAMETER ESTIMATION 5-12 Estimating parameters (Quinn and Keough 2002)
14 CHAPTER 5: PARAMETER ESTIMATION 5-13 Estimating parameters
15 CHAPTER 5: PARAMETER ESTIMATION 5-14 Using statistics (derived from samples) to do the job
16 CHAPTER 5: PARAMETER ESTIMATION 5-15 Point and interval estimates (Quinn and Keough 2002)
17 CHAPTER 5: PARAMETER ESTIMATION 5-16 Testing hypotheses Chapter 6 Is it safe to say that you accept a null hypothesis? (Quinn and Keough 2002)
18 CHAPTER 5: PARAMETER ESTIMATION 5-17 There are several inferential frameworks Each of these offer their own machinery and set of tools to make inferences Recall (Chapter 4): o For classical (traditional) analysis, the sequence is Problem => Data => Model => Analysis => Conclusions o For EDA, the sequence is Problem => Data => Analysis => Model => Conclusions o For Bayesian, the sequence is Problem => Data => Model => Prior Distribution => Analysis => Conclusions
19 CHAPTER 5: PARAMETER ESTIMATION 5-18 Inferential framework 1: parametric (frequentist) analysis The data collection is followed by the imposition of a model (normality, linearity, etc.) and the analysis, estimation, and testing that follows are focused on the parameters of that model.
20 CHAPTER 5: PARAMETER ESTIMATION 5-19 Inferential framework 1: parametric (frequentist) analysis (Quinn and Keough 2002) How would you get an idea about which model(s) to impose?
21 CHAPTER 5: PARAMETER ESTIMATION 5-20 Inferential framework 1: Bayesian analysis For a Bayesian analysis, the analyst attempts to incorporate scientific/engineering knowledge/expertise into the analysis by imposing a data-independent distribution on the parameters of the selected model; the analysis thus consists of formally combining both the prior distribution on the parameters and the collected data to jointly make inferences and/or test assumptions about the model parameters. Or formulated differently (making links to Bayesian theorem and conditionality)
22 CHAPTER 5: PARAMETER ESTIMATION 5-21 Inferential framework 1: Bayesian analysis (Quinn and Keough 2002)
23 CHAPTER 5: PARAMETER ESTIMATION 5-22 Inferential framework 3: non-parametric analysis and others
24 CHAPTER 5: PARAMETER ESTIMATION 5-23 Bootstrapping Monte Carlo (resampling) analysis (Quinn and Keough 2002)
25 CHAPTER 5: PARAMETER ESTIMATION Estimation by the Method of Moments
26 CHAPTER 5: PARAMETER ESTIMATION 5-25
27 CHAPTER 5: PARAMETER ESTIMATION 5-26 Example 1
28 CHAPTER 5: PARAMETER ESTIMATION 5-27 Example 2
29 CHAPTER 5: PARAMETER ESTIMATION 5-28 General procedure
30 CHAPTER 5: PARAMETER ESTIMATION Estimation by the Method of Maximum Likelihood Likelihood of a sample
31 CHAPTER 5: PARAMETER ESTIMATION 5-30 Likelihood of a sample
32 CHAPTER 5: PARAMETER ESTIMATION 5-31 Likelihood of a sample (Quinn and Keough 2002)
33 CHAPTER 5: PARAMETER ESTIMATION 5-32 The likelihood function for estimating the probability of a coin landing headsup without prior knowledge after observing HH
34 CHAPTER 5: PARAMETER ESTIMATION 5-33 The likelihood function for estimating the probability of a coin landing headsup without prior knowledge after observing HHT
35 CHAPTER 5: PARAMETER ESTIMATION 5-34 Maximum likelihood estimators (less reliable for small sample sizes and unusual distributions)
36 CHAPTER 5: PARAMETER ESTIMATION 5-35 Maximum likelihood estimators
37 CHAPTER 5: PARAMETER ESTIMATION 5-36 Maximum likelihood estimators: some remarks
38 CHAPTER 5: PARAMETER ESTIMATION 5-37 Example 1
39 CHAPTER 5: PARAMETER ESTIMATION 5-38
40 CHAPTER 5: PARAMETER ESTIMATION 5-39 Example 2
41 CHAPTER 5: PARAMETER ESTIMATION 5-40
42 CHAPTER 5: PARAMETER ESTIMATION 5-41 Example 3
43 CHAPTER 5: PARAMETER ESTIMATION 5-42 Example 3 continued
44 CHAPTER 5: PARAMETER ESTIMATION 5-43 Ordinary least squares estimators OLS estimators minimize the sum of squared deviations from the trendline (hence between each observation and estimated value) 1-r 2 = (deviations from trendline) (standard deviation of y data) (reliable for linear models with normal distributions)
45 CHAPTER 5: PARAMETER ESTIMATION 5-44 Example
46 CHAPTER 5: PARAMETER ESTIMATION 5-45 Example
47 CHAPTER 5: PARAMETER ESTIMATION Properties of Estimators 2.1 Introduction
48 CHAPTER 5: PARAMETER ESTIMATION Unbiased
49 CHAPTER 5: PARAMETER ESTIMATION Efficiency
50 CHAPTER 5: PARAMETER ESTIMATION 5-49 Practical The MMEs give unbiased estimates which may or may not be in the range space. The MLEs are all less than 10 and hence biased. What if the sample size is increased? (think about CLT)
51 CHAPTER 5: PARAMETER ESTIMATION Consistency
52 CHAPTER 5: PARAMETER ESTIMATION 5-51 Estimator MLE Property Unbiased Efficient Consistent Invariant They become They become minimum minimum variance variance unbiased unbiased estimators as estimators as the sample size the sample size increases increases MLEs are invariant with respect to reparametrisations : if you have found the MLE for a parameter, then you have found one for, namely the g(mle), where g is an invertible function MLEs have approximate normal distributions and approximate sample variances that can be used to generate confidence bounds and hypothesis tests for the parameters.
53 CHAPTER 5: PARAMETER ESTIMATION Confidence intervals 3.1 Introduction understanding the concept From point estimators to interval estimators
54 CHAPTER 5: PARAMETER ESTIMATION 5-53 Example
55 CHAPTER 5: PARAMETER ESTIMATION 5-54.
56 CHAPTER 5: PARAMETER ESTIMATION 5-55 With
57 CHAPTER 5: PARAMETER ESTIMATION 5-56 Correct interpretation of probabilities associated to intervals
58 CHAPTER 5: PARAMETER ESTIMATION 5-57 The vertical line segments represent 50 realizations of a confidence interval for μ.
59 CHAPTER 5: PARAMETER ESTIMATION 5-58
60 CHAPTER 5: PARAMETER ESTIMATION 5-59 Confidence intervals are not unique
61 CHAPTER 5: PARAMETER ESTIMATION Finding confidence intervals in practice A pivotal quantity or pivot is generally defined as a function of observations and unobservable parameters whose probability distribution does not depend on unknown parameters Can we use pivots in constructing confidence intervals?
62 CHAPTER 5: PARAMETER ESTIMATION 5-61 The pivotal method
63 CHAPTER 5: PARAMETER ESTIMATION 5-62 Example
64 CHAPTER 5: PARAMETER ESTIMATION One-sample problem Confidence Intervals for (memorize) Case (i) Case (ii)
65 CHAPTER 5: PARAMETER ESTIMATION 5-64 Case (i)
66 CHAPTER 5: PARAMETER ESTIMATION 5-65 Case (ii)
67 CHAPTER 5: PARAMETER ESTIMATION 5-66
68 CHAPTER 5: PARAMETER ESTIMATION 5-67 Confidence Intervals for (memorize) Population Estimation of Test statistic and distribution Case (i) Known Case (ii) Unknown unbiased for,
69 CHAPTER 5: PARAMETER ESTIMATION 5-68 Note For Z and W defined as above, Z ~ N(0,1) and. So according to the definition of a t-distribution: Now,
70 CHAPTER 5: PARAMETER ESTIMATION 5-69 Case (ii)
71 CHAPTER 5: PARAMETER ESTIMATION 5-70
72 CHAPTER 5: PARAMETER ESTIMATION 5-71 Remark
73 CHAPTER 5: PARAMETER ESTIMATION 5-72 Case (i)
74 CHAPTER 5: PARAMETER ESTIMATION Two-sample problems Confidence Interval for (note that nothing is said about the means) F -distribution
75 CHAPTER 5: PARAMETER ESTIMATION 5-74
76 CHAPTER 5: PARAMETER ESTIMATION 5-75 Confidence intervals for means in different populations (memorize) START
77 CHAPTER 5: PARAMETER ESTIMATION 5-76 Pooling variances
78 CHAPTER 5: PARAMETER ESTIMATION 5-77 Do you know why?
79 CHAPTER 5: PARAMETER ESTIMATION 5-78 Confidence Interval for (known variances) (memorize)
80 CHAPTER 5: PARAMETER ESTIMATION 5-79 Confidence Interval for (unknown variances) Either the population variances are unequal Either the population variances are equal o If it is reasonable to assume that σ 1 =σ 2, we can estimate the standard error more efficiently by combining the sample. o Assume equal variances when S 2 /S 1 < 2 o If you are unsure, the unequal variance formula will be the conservative choice (less power, but less likely to be incorrect).
81 CHAPTER 5: PARAMETER ESTIMATION 5-80 With the pooled variance, the key expression to remember is with the pooled variance as described on the previous slide, naturally leading to a percent confidence interval.
82 CHAPTER 5: PARAMETER ESTIMATION 5-81 Main reference: STAT261 Statistical inference notes School of mathematics, statistics and computer science. University of New England, Oct 4, 2007
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