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Power Calculation
Creation date：2024-05-10 15:00:34
Belonging category：数学mathematics
Power Calculation

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Power Calculation

• Suppose, for illustation, that we are interested in testing the hypothesis
H0: ✓1 = ✓2 vs. HA: ✓1 6= ✓2
• Suppose, also for illustration, that the test statistic associated with this test has the form
T =
(Y 1 Y 2) (✓1 ✓2)q
Var[Y1]
n +
Var[Y2]
n
⇠ N(0, 1)
• It will be useful to define the notion of a rejection region R: all values of the observed test statistic
t that would lead to the rejection of H0:
R = {t | H0 is rejected}
– If t 2 R, we reject H0
– If t 2 Rc, we do not reject H0
1

• Defining Type I and Type II error rates in terms of a rejection region is also useful:
↵ = Pr(Type I Error) = Pr(Reject H0 | H0 is true)
= Pr(T 2 R | H0 is true)
= Pr(Type II Error) = Pr(Do Not Reject H0 | H0 is false)
= Pr(T 2 Rc | H0 is false)
2
3
Permutation and Randomization Tests
• All of the previous tests have made some kind of distributional assumption for the response measure-
ments
• It would be preferable to have a test that does not rely on any assumptions
• This is precisely the purpose of permutation and randomization tests.
– These tests are nonparametric and rely on resampling.
– The motivation is that if H0 : ✓1 = ✓2 is true, any random rearrangement of the data is equally
likely to have been observed.
– With n1 and n2 units in each condition, there are✓
n1 + n2
n1
◆
=
✓
n1 + n2
n2
◆
arrangements of the n1 + n2 observations into two groups of size n1 and n2 respectively
4
• A true permutation test considers all possible rearrangements of the original data
– The test statistic t is calculated on the original data and on every one of its rearrangements
– This collection of test statistic values generate the empirical null distribution
• A randomization test is carried out similarly, except that we do not consider all possible rearrange-
ments
– We just consider a large number N of them
Randomization Test Algorithm
1. Collect response observations in each condition.
2. Calculate the test statistic t on the original data.
5
3. Pool all of the observations together and randomly sample (without replacement) n1 observations which
will be assigned to “Condition 1” and the remaining n2 observations are assigned to “Condition 2”.
Repeat this N times.
4. Calculate the test statistic t?k on each of the “shu✏ed” datasets, k = 1, 2, . . . , N .
5. Compare t to {t?1, t?2, . . . , t?N}, the empirical null distribution and calculate the p-value:
p-value =
# of t?’s that are at least as extreme as t
N
Example: Pokemon Go
• Suppose that Niantic is experimenting with two di↵erent promotions within Poke´mon Go:
– Condition 1: Give users nothing
– Condition 2: Give users 200 free Poke´coins
– Condition 3: Give users a 50% discount on Shop purchases
• In a small pilot experiment n1 = n2 = n3 = 100 users are randomized to each condition
• For each user, the amount of real money (in USD) they spend in the 30 days following the experiment
is recorded
• The data summaries are:
– y1 = $10.74, Q1(0.5) = $9
– y2 = $9.53, Q2(0.5) = $8
– y3 = $13.41, Q3(0.5) = $10
6
3 Experiments with More than Two Conditions
3.1 Anatomy of an A/B/m Test
• We now consider the design and analysis of an experiment consisting of more than two experimental
conditions – or what many data scientists broadly refer to as “A/B/m Testing”.
– Canonical A/B/m test:
Figure 1: Button-Colour Experiment
• Other, more tangible, examples:
– Netflix
– Etsy
• Typically the goal of such an experiment is to decide which condition is optimal with respect to some
metric of interest ✓. This could be a
– mean
– proportion
– variance
– quantile
– technically any statistic that can be calculated from sample data
• From a design standpoint, such an experiment is very similar to a two-condition experiment
1. Choose a metric of interest ✓ which addresses the question you are trying to answer
2. Determine the response variable y that must be measured on each unit in order to estimate b✓
3. Choose the design factor x and the m levels you will experiment with.
4. Choose n1, n2, . . . , nm and assign units to conditions at random
5. Collect the data and estimate the metric of interest in each condition:
b✓1, b✓2, . . . , b✓m
7
• Determining which condition is optimal typically involves a series of pairwise comparisons
• But it is useful to begin such an investigation with a gatekeeper test which serves to determine whether
there is any di↵erence between the m experimental conditions. Formally, such a question is phrased
as the following statistical hypothesis.
H0: ✓1 = ✓2 = · · · = ✓m versus HA: ✓j 6= ✓k for some j 6= k (1)
3.2 Comparing Multiple Means with an F -test
• We assume that our response variable follows a normal distribution and we assume that the mean of
the distribution depends on the condition in which the measurements were taken, and that the variance
is the same across all conditions.
• The “gatekeeper” test for means is tested using an F -test
• In particular, we use the F -test for overall significance in an appropriately defined linear regression
model :
– The appropriately defined linear regression model in this situation is one in which the response
variable depends on m 1 indicator variables:
xij =
(
1 if unit i is in condition j
0 otherwise
for j = 1, 2, . . . ,m 1.
– For a particular unit i, we adopt the model
Yi = 0 + 1xi1 + 2xi2 + · · ·+ m1xi,m1 + "i
8
– In this model the ’s are unknown parameters and may be interpreted in the context of the
following expectations:
E[Yi|xi1 = xi2 = · · · = xi,m1 = 0] = 0
E[Yi|xij = 1] = 0 + j
– Based on these assumptions, H0 in (1) is true if and only if 1 = 2 = · · · = m1 = 0. Thus
testing (1) is equivalent to testing
H0: 1 = 2 = · · · = m1 = 0 vs. HA: j 6= 0 for some j
– This hypothesis corresponds, as noted, to the F -test for overall significance in the model.
• In regression parlance, the test statistic is defined to be the ratio of the regression mean squares (MSR)
to the mean squared error (MSE) in a standard regression-based analysis of variance (ANOVA):
t =
MSR
MSE
• In our setting we can more intuitively think of the test statistic as comparing the response variability
between conditions to the response variability within conditions:
9
• The null distribution for this test is F(m1,Nm)
• The p-value for this test is calculated by
p-value = P (T t)
where T ⇠ F(m1,Nm)
• Example: Candy Crush Boosters
– Candy Crush is experimenting with three di↵erent versions of in-game “boosters”: the lollipop
hammer, the jelly fish, and the color bomb.
Figure 2: Candy Crush Experiment
– Users are randomized to one of these three conditions (n1 = 121, n2 = 135, n3 = 117) and they
receive (for free) 5 boosters corresponding to their condition. Interest lies in evaluating the e↵ect
of these di↵erent boosters on the length of time a user plays the game.
– Let µj represent the average length of game play (in minutes) associated with booster condition
j = 1, 2, 3. While interest lies in finding the condition associated with the longest average length
of game play, here we first rule out the possibility that booster type does not influence the length
of game play (i.e., µ1 = µ2 = µ3).
– In order to do this we fit the linear regression model
Y = 0 + 1x1 + 2x2 + "
where x1 and x2 are indicator variables indicating whether a particular value of the response was
observed in the jelly fish or color bomb conditions, respectively. The lollipop hammer is therefore
the reference condition.
10
Optional Exercises:
• Calculations: 2, 7
• Proofs: 1, 5, 6, 9, 10, 14, 17, 18
• R Analysis: 2, 5, 6, 8, 13(g), 17 (not g,h), 22(h), 23(a-f)
Case category
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CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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