LECTURE NOTES Repeated Measures Analysis: MANOVA and Covariance Pattern models

1
LECTURE NOTESRepeated Measures Analysis
MANOVA and Covariance Pattern models
2
Day16 Basic Repeated Measures Design

• Data collected in a sequence of evenly spaced
  points in time (not necessarily equally spaced)
• Treatments are assigned to experimental units
• I.e., subjects
• Two factors
• Treatment between-subjects factor
• Time within-subjects factor

3
Hypotheses

• How do treatment differences change over time?
• Is there a Treatment ? Time interaction?
• How do response means change by trt?
• Is there a Trt main effect?
• How do response means change over time?
• Is there a Time main effect?

4
Example Two groups
id group time1 time2 time3 time4 1
A 31 29 15 26 2
A 24 28 20 32 3
A 14 20 28 30 4
B 38 34 30 34 5
B 25 29 25 29 6
B 30 28 16 34

• Preliminary Analysis this includes
• Profile plots
• Mean plots
• Correlation between repeated measurements

5
Profile plots by group

• differences at baseline among subjects
• different trends for different subjects
• Variability higher at time 1 and low at time 4

B
A
6
Mean plots by group

• differences at baseline between group means
• non-linear trends

B
A
7
(b) Correlation (covariance) across time points
time1 time2 time3
time4 time1 1.00000 0.94035
-0.14150 0.28445 time2 0.94035
1.00000 -0.02819 0.26921
time3 -0.14150 -0.02819
1.00000 0.27844 time4 0.28445
0.26921 0.27844 1.00000
Certainly do NOT have equal correlations (CS?)!
Time1 and time2 are highly correlated, but time1
and time3 are inversely correlated!
8
Statistical analysis strategies

• Strategy 1 ANCOVA on the final measurement,
  adjusting for baseline differences (end-point
  analysis)
• Strategy 2 repeated-measures ANOVA
  Univariate approach
• Strategy 4 Multivariate ANOVA approach
• Strategy 3 Summary approach
• Strategy 5 GEE
• Strategy 6 Mixed Models

9
Comparison of traditional and new methods
FROM Ralitza Gueorguieva, PhD John H. Krystal,
MD Move Over ANOVA Progress in Analyzing
Repeated-Measures Data and Its Reflection in
Papers Published in the Archives of General
Psychiatry. Arch Gen Psychiatry. 200461310-317.
10
General syntax in SAS's GLM procedure

• Syntax for MANOVA and rANOVA
• PROC GLM DATA sas-dataset-name   CLASS
  factor1 factor2 ... factork   MODEL y1 y2 ...
  yk factor1 ... factork   REPEATED
  repeated-factor-name k / PRINTE   LSMEANS
  factor1 factor2 ... factork RUN
• Output can be restricted to rANOVA only using NOM
  option and to MANOVA analysis using NOU in the
  Repeated Statement

11
Strategy 1. End-point analysis
ANCOVA Asks whether or not the two group means
differ at the final time point, adjusting for
differences at baseline (using ANCOVA).
proc glm datahorizontal class group model
time4 time1 group run
- Comparing groups at every follow-up time point
in this way would hugely increase your type I
error.
12
Strategy 2 univariate repeated measures ANOVA
(rANOVA)
Explain away some error variability by accounting
for differences between subjects - requires
Sphericity
proc glm datahorizontal class group
model time1-time4 group repeated time
4/printe run quit
13
Strategy 3 Summary analysis

• One way to overcome the problem of correlated
  observations over time within each subject is to
  summarize the observations over time by their
  mean or some function and use ANOVA
• This summary analysis leads to a conservative
  test
• Example, avetimemean(time1,time2,time3,time4)

proc glm datahorizontal class group
model avetime group run quit
- A special application of this is pre-post
analysis
14
Strategy4 MANOVA Approach

• Successive response measurements made over time
  are considered correlated dependent variables
• That is, response variables for each level of
  within-subject factor is presumed to be a
  different dependent variable
• MANOVA assumes there is an unstructured
  covariance matrix for dependent variables

15
Why MANOVA

• You do a MANOVA instead of a series of
  one-at-a-time ANOVAs for two main reasons
• to reduce the experiment-wise level of Type I
  error.
• None of the individual ANOVAs may produce a
  significant main effect on the response, but in
  combination they might, which suggests that the
  variables are more meaningful taken together than
  considered separately
• MANOVA takes into account the inter-correlations
  among the response Variables

16
MANOVA

• If the multivariate test is
• not significant, report no group differences
  among the mean vectors
• significant, perform univariate ANOVA and
  relevant contrasts
• Contrasts (similar to contrasts we considered
  previously)
• Prior (planned)
• Post hoc (unplanned)

17
MANOVA Test Statistics

• SAS reports four tests
• Wilks Lmbda
• Pillais trace (good for smaller sample size)
• Hotelling- Lawley Trace
• Roys greatest root
• These are covered in Multivariate class
• We will use results from Wilks

18
MANOVA Test Statistics

• Wilks Lambda (?) was the first MANOVA test
  statistic developed and is very important for
  several multivariate procedures in addition to
  MANOVA.
• Wilks Lambda (?) is the error sum of squares (E)
  divided by the sum of the effect sum of squares
  (H) and the error sum of squares.
• The quantity (1 - ?) is often interpreted as the
  proportion of variance in the dependent variables
  explained by the model effect. However, this
  quantity is not unbiased and can be quite
  misleading in small samples.
• ? is approximately chi-square distributed

19
rANOVA vs. MANOVA

• For tests that involve only between-subjects
  effects, both the MANOVA rANOVA give rise to
  the same tests.
• For within-subject effects they yield different
  tests.
• In Proc GLM, rANOVA are in a table "Univariate
  Tests of Hypotheses for Within Subject Effects."
• Results for MANOVA are displayed in a table
  labeled "Repeated Measures Analysis of Variance.
• The multivariate tests are Wilks lambda,
  Pillais trace, Hotelling-Lawley trace, and Roys
  greatest root.
• The only assumption required for valid tests is
  that the dependent variables in the model have a
  multivariate normal distribution with a common
  covariance matrix across the between-subject
  effects.

20
Boxs test of equal covariances

• Boxs M test can be used if there are significant
  differences among the covariance matrices by
  group.
• when Boxs test finds that the covariance
  matrices are significantly different across
  groups that may indicate an increased possibility
  of Type I error, so you might want to make a
  smaller error region (alpha0.001).
• If you redid the analysis with a confidence level
  of .001, you should report the results of the
  Boxs M.

Box's M test for equality of variances proc
discrim dataexercise methodnormal pooltest
wcov class diet var time1 time2 time3 run
21

• Example3 Suppose 24 subjects are randomly
  assigned to two groups (Control and Treatment)
  and their responses are measured at 4 times.
  These times are labeled as 0 (baseline), 1 (after
  one month posttest) 3 (after 3 months of
  follow-up) and 6 (after 6 months of follow-up).
• time is the within-subjects factor in this design
  
• Treatment is the between-subjects (grouping)
  factor

22

• Some of the data points
• data short
• input Group Subj y0 y1 y3 y6
• datalines
• 1 1 296 175 187 242
• 1 2 376 329 236 126
• 1 3 309 238 150 173
• 1 4 222 60 82 135
• 1 5 150 271 250 266
• 1 6 316 291 238 194
• 1 7 321 364 270 358
• 1 8 447 402 294 266
• 1 9 220 70 95 137
• 2 23 319 68 67 12
• 2 24 300 138 114 12

Hypothesis H01 no trt effect H02 no time
effect H03 no interaction Sphericity is
violated
23

Results of GLM Analysis
24

Results of GLM Analysis
25

Results of GLM Analysis

• The test of sphericity, when requested,
  immediately precedes both sets of within-subjects
  tests.
• Although the output shows two separate tests of
  sphericity, the only one of interest is the
  second test, which is the test of sphericity
  applied to the common covariance matrix of the
  transformed within-subject variables.
• If the Chi-square approximation has an associated
  p value less than your alpha level, the
  sphericity assumption has been violated

Sphericity Tests Sphericity Tests Sphericity Tests Sphericity Tests Sphericity Tests
Variables DF Mauchly's Criterion Chi-Square Pr gt ChiSq
Transformed Variates 5 0.462959 15.95853 0.0070
Orthogonal Components 5 0.462959 15.95853 0.0070
26

Example 3 continued rMANOVA

• The first multivariate test of a within-subjects
  effect is the within-subjects main effect test.
• It examines changes in response as a function of
  time.
• The null hypothesis is that the mean response
  does not change over time.

MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no time EffectH Type III SSCP Matrix for timeE Error SSCP MatrixS1 M0.5 N9 MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no time EffectH Type III SSCP Matrix for timeE Error SSCP MatrixS1 M0.5 N9 MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no time EffectH Type III SSCP Matrix for timeE Error SSCP MatrixS1 M0.5 N9 MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no time EffectH Type III SSCP Matrix for timeE Error SSCP MatrixS1 M0.5 N9 MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no time EffectH Type III SSCP Matrix for timeE Error SSCP MatrixS1 M0.5 N9 MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no time EffectH Type III SSCP Matrix for timeE Error SSCP MatrixS1 M0.5 N9
Statistic Value F Value Num DF Den DF Pr gt F
Wilks' Lambda 0.19328615 27.82 3 20 lt.0001
Pillai's Trace 0.80671385 27.82 3 20 lt.0001
Hotelling-Lawley Trace 4.17367645 27.82 3 20 lt.0001
Roy's Greatest Root 4.17367645 27.82 3 20 lt.0001
27

• Next SAS tests the hypothesis that treatment
  interacts with time.
• In this instance, the F value associated with
  these multivariate tests of the interaction is
  high therefore, the associated p value is low
  F(3, 20) 6.73, p .0025.

MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no timeGroup EffectH Type III SSCP Matrix for timeGroupE Error SSCP MatrixS1 M0.5 N9 MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no timeGroup EffectH Type III SSCP Matrix for timeGroupE Error SSCP MatrixS1 M0.5 N9 MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no timeGroup EffectH Type III SSCP Matrix for timeGroupE Error SSCP MatrixS1 M0.5 N9 MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no timeGroup EffectH Type III SSCP Matrix for timeGroupE Error SSCP MatrixS1 M0.5 N9 MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no timeGroup EffectH Type III SSCP Matrix for timeGroupE Error SSCP MatrixS1 M0.5 N9 MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no timeGroup EffectH Type III SSCP Matrix for timeGroupE Error SSCP MatrixS1 M0.5 N9
Statistic Value F Value Num DF Den DF Pr gt F
Wilks' Lambda 0.49748100 6.73 3 20 0.0025
Pillai's Trace 0.50251900 6.73 3 20 0.0025
Hotelling-Lawley Trace 1.01012703 6.73 3 20 0.0025
Roy's Greatest Root 1.01012703 6.73 3 20 0.0025
28

• Between-Subjects Tests
• Following the MANOVA for multivariate tests of
  significance for within-subjects effects, SAS
  prints tests of the between-subjects effects.
  There is only one approach to testing these
  effects.

The GLM Procedure Repeated Measures Analysis of
Variance Tests of Hypotheses for Between Subjects
Effects
Source DF Type III SS Mean Square F Value Pr gt F
Group 1 248677.0417 248677.0417 19.64 0.0002
Error 22 278540.9583 12660.9527
Source DF Type III SS Mean Square F Value Pr gt F Adj Pr gt F Adj Pr gt F
Source DF Type III SS Mean Square F Value Pr gt F G - G H-F-L
time 3 326635.5833 108878.5278 37.80 lt.0001 lt.0001 lt.0001
timeGroup 3 59461.8750 19820.6250 6.88 0.0004 0.0019 0.0012
Error(time) 66 190098.5417 2880.2809
Greenhouse-Geisser Epsilon 0.7204
Huynh-Feldt-Lecoutre Epsilon 0.8016
29

• Observations
• The sphericity assumption was violated
• With nonspherical data either use corrected
  univariate tests that we described earlier or use
  results from MANOVA test.
• The corrected univariate p values appear under
  the G - G and H - F headers in the output shown
  above.
• Note that in this case, rANOVA agrees with the
  MANOVA that there is a statistically significant
  within-subjects main effect for time, as well as
  interaction between treatment and time.
• Further polynomial contrast analysis can be made
  on time

30
Analysis of Variance of Contrast Variables time_N
represents the nth degree polynomial contrast for
time
Source DF Type III SS Mean Square F Value Pr gt F
Mean 1 89560.72781 89560.72781 44.23 lt.0001
Group 1 20747.22078 20747.22078 10.24 0.0041
Error 22 44552.41504 2025.10977
time_1
time_2
Source DF Type III SS Mean Square F Value Pr gt F
Mean 1 186802.0020 186802.0020 37.82 lt.0001
Group 1 4428.6429 4428.6429 0.90 0.3539
Error 22 108650.6885 4938.6677
time_3
Source DF Type III SS Mean Square F Value Pr gt F
Mean 1 50272.85354 50272.85354 29.98 lt.0001
Group 1 34286.01136 34286.01136 20.44 0.0002
Error 22 36895.43813 1677.06537
31
More on orthogonal Contrast

• proc glm datashort
• class group
• model y0 y1 y3 y6 group/ nouni
• repeated time 4 (0 1 3 6) profile /summary
  printm NOM generates contrasts between adjacent
  levels of the factor
• proc glm datashort
• class group
• model y0 y1 y3 y6 group/ nouni
• repeated time 4 (0 1 3 6) helmert /summary
  printm NOM HELMERT-generates contrasts between
  each level of the factor and the mean of
  subsequent levels.
• run

32
Day17 Strategy 6 Mixed Model Approach

• Models with fixed and random effects are mixed
  models
• treatment, which is usually considered a fixed
  effect
• subject factor is a random effect
• Analysis can follow
• Linear Mixed models
• Covariance pattern models user specifies
  covariance structure
• Random coefficient models induce covariance
  structure

33
SAS Mixed Repeated Measures Syntax
34
SAS Mixed Model

• PROC MIXED cl
• CLASS
• MODEL ltdependent variablegt ltfixed sourcesgt

• cl requests confidence limits for variance
  covariance estimates
• Identifies variables used as sources of variation
  and subject option of REPEATED statement
• Specifies dependent variable and all fixed
  sources of variation (includes treatment, time
  and their interaction. The ddfm option computes
  the correct degrees of freedom for the various
  terms.

35
SAS Mixed Model

• REPEATED/ subject ltEU idgt typeltcovariance
  structuregt r rcorr

• subject identifies the experimental unit in
  the data set which represents the repeated
  measures. It identifies the units that are
  indpendent.
• type identifies the covariance structure
• r requests printing of the covariance matrix for
  the repeated measures
• rcorr requests printing of the correlation matrix
  for the repeated measures

36
Covariance Structures Independent with common
variance

• Equal variances along main diagonal
• Zero covariances along off diagonal
• Variances constant and residuals independent
  across time.
• The standard ANOVA model
• Simple, because a single parameter is estimated
  the pooled variance

37
Covariance Structures Unstructured

• Separate variances on diagonal
• Separate covariances on off diagonal
• Most complex structure
• Variance estimated for each time, covariance for
  each pair of times
• Need to estimate 10 parameters, 104(41)/2
• Leads to less precise parameter estimation
  (degrees of freedom problem)

38
Covariance Structures compound symmetry

• Equal variances on diagonal
• equal covariances along off diagonal (equal
  correlation)
• Simplest structure for fitting repeated measures
• Split-plot in time analysis
• Used for past 50 years
• Requires estimation of 2 parameters

39
Covariance Structures First order
Autoregressive

• Equal variances on main-diagonal
• Off diagonal represents variance multiplied by
  the correlation raised to increasing powers as
  the observations become increasingly separated in
  time.
• Increasing power means decreasing covariances.
• Times must be equally ordered and equally spaced.
• Estimates 2 parameters
• AR(1)

40
First order Autoregressive Heterogeneous

• unequal variances on main-diagonal
• Off diagonal represents product of standard
  errors multiplied by the correlation raised to
  increasing powers as the observations become
  increasingly separated in time.
• Increasing power means decreasing covariances.
• Times must be equally ordered and equally spaced.
• Estimates 5 parameters
• ARH(1)

41
Strategies for Finding suitable covariance
structures

• Run unstructured first
• Next run compound symmetry simplest repeated
  measures structure
• Next try other covariance structures that best
  fit the experimental design

42
Criteria for Selecting best Covariance Structure

• Need to use model fitting statistics
• AIC Akaikes Information Criteria
• BIC Schwarzs Bayesian Criteria
• Let q of covariance parameters, p of fixed
  effect parameters in model and n of
  observations and
• AIC -2log(L) 2q
• BIC -2log(L) q log(n)
• AAIC -2log(L) q(logn 1)
• Smaller the number the better
• Goal covariance structure that is better than
  compound symmetry

43

• Example3 Suppose 24 subjects are randomly
  assigned to two groups (Control and Treatment)
  and their responses are measured at 4 times.
  These times are labeled as 0 (baseline), 1 (after
  one month posttest) 3 (after 3 months of
  follow-up) and 6 (after 6 months of follow-up).

proc corr datashort cov var yo y1 y3
y6 run Pearson Correlation Coefficients, N
24 yo y1 y3 y6 Y0 1.00
0.51 0.50 0.07 y1 1.00
0.93 0.67 y3 1.00 0.65 Y4 1.
00

• - What type of correlation structure do you think
  is right?
• Variances are 5456, 13505, 7881,6929
• exercise compare models for this

44
Example4 exercise pulse study

• Exercise data examples
• The data consists of people who were randomly
  assigned to two different diets low-fat and not
  low-fat and three different types of exercise at
  rest, walking leisurely and running. Their pulse
  rate was measured at three different time points
  during their assigned exercise at 1 minute, 15
  minutes and 30 minutes.
• data exercise
• input id exertype diet time1 time2 time3
• cards
• 1 1 1 85 85 88
• 2 1 1 90 92 93
• 3 1 1 97 97 94
• 4 1 1 80 82 83
• 5 1 1 91 92 91
• 6 1 2 83 83 84
• 7 1 2 87 88 90
• 8 1 2 92 94 95
• 9 1 2 97 99 96
• 10 1 2 100 97 100

45
Example

• Let's look at the correlations, variances and
  covariances for the exercise data.
• since, we cannot use this kind of covariance
  structures in a traditional repeated measures
  analysis, we will use SAS PROC MIXED for such an
  analysis.
• proc corr dataexercise cov
• var time1 time2 time3
• run
• Pearson Correlation Coefficients, N 30
• time1 time2 time3
• time1 1.00000 0.54454 0.51915
• time2 0.54454 1.00000 0.85028
• time3 0.51915 0.85028 1.00000

46
Example compound symmetry

• proc mixed datalong
•   class exertype time
•   model pulse exertype time exertypetime
•   repeated time / subjectid typecs
• run
• Fit Statistics
• -2 Res Log Likelihood 590.8
• AIC (smaller is better) 594.8
• AICC (smaller is better) 595.0
• BIC (smaller is better) 597.6
• Null Model Likelihood Ratio Test
• DF Chi-Square Pr gt ChiSq
• 1 15.36 lt.0001
• Type 3 Tests of Fixed Effects
• Num Den
• Effect DF DF F Value Pr gt
  F
• exertype 2 27 27.00
  lt.0001
• time 2 54 23.54
  lt.0001

47
Example unstructured

• proc mixed datalong
•   class exertype time
•   model pulse exertype time exertypetime
•   repeated time / subjectid typeun
• run
• Fit Statistics
• -2 Res Log Likelihood 577.7
• AIC (smaller is better) 589.7
• AICC (smaller is better) 590.9
• BIC (smaller is better) 598.1
• Null Model Likelihood Ratio Test
• DF Chi-Square Pr gt ChiSq
• 5 28.46 lt.0001
• Type 3 Tests of Fixed Effects
• Num Den
• Effect DF DF F Value Pr gt
  F
• exertype 2 27 27.00
  lt.0001
• time 2 27 22.32
  lt.0001

48
Example AR(1)

• proc mixed datalong
•   class exertype time
•   model pulse exertype time exertypetime
•   repeated time / subjectid typear(1)
• run
• -2 Res Log Likelihood 590.1
• AIC (smaller is better) 594.1
• AICC (smaller is better) 594.3
• BIC (smaller is better) 596.9
• Null Model Likelihood Ratio Test
• DF Chi-Square Pr gt ChiSq
• 1 16.08 lt.0001
• Type 3 Tests of Fixed Effects
• Num Den
• Effect DF DF F Value Pr gt
  F
• exertype 2 27 28.39
  lt.0001
• time 2 54 18.20
  lt.0001
• exertypetime 4 54 11.73 lt.0001

49
Example ARH(1)

• proc mixed datalong
•   class exertype time
•   model pulse exertype time exertypetime
•   repeated time / subjectid typearh(1)
• run
• Covariance Parameter Estimates
• Cov
• Parm Subject Estimate
• Var(1) id 35.7683
• Var(2) id 87.1927
• Var(3) id 115.50
• ARH(1) id 0.5101
• Fit Statistics
• -2 Res Log Likelihood 579.8
• AIC (smaller is better) 587.8
• AICC (smaller is better) 588.3
• BIC (smaller is better) 593.4
• Null Model Likelihood Ratio Test
• DF Chi-Square Pr gt ChiSq

50
Example model comparison
Model AIC -2RLL Parms(df 1) Diff -2RLL(vs. CS) Diff in df (vs. CS) p value for Diff (from a chi square dist)
Compound Symmetry 594.8 590.8 2      
Unstructured 589.7 577.7 6 13.1 4 .01
Autoregressive 594.1 590.1 2 .7 0 na
Autoregressive Heterogenous Variances 587.8 579.8 4 11 2 0.027
The two most promising structures are
Autoregressive Heterogeneous Variances and
Unstructured since these two models have the
smallest AIC values and the -2 Log Likelihood
scores are significantly smaller than the -2 Log
Likehood scores of other models.
51
RM with two group factors

• Looking at models including only diet or exertype
  separately does not answer all our questions. We
  would also like to know if the people on the
  low-fat diet who engage in running have lower
  pulse rates than the people participating in the
  not low-fat diet who are not running. In order to
  address these types of questions we need to look
  at a model that includes the interaction of diet
  and exertype.
• proc mixed datalong
• class diet exertype time
• model pulse exertypediettime
• repeated time / subjectid typearh(1)
• run
• quit
• proc glm dataexercise
• class diet exertype
• model time1 time2 time3 dietexertype
• repeated time 3
• run
• quit

52
Group comparison in Proc Mixed

• If we would like to look at the differences among
  groups at each level of another variable we have
  to utilize the lsmeans statement with the slice
  option.
• For example, we could test for differences among
  the exertype groups at each level of diet across
  all levels of time or we could test for
  differences in groups of exertype for each time
  point across both levels of diet we could also
  test for differences in groups of exertype for
  each combination of time and diet levels.
• proc mixed datalong
• class diet exertype time
• model pulse exertypediettime
• repeated time / subjectid typearh(1)
• lsmeans dietexertype / slicediet /testing
  for differences among exertype for each level of
  diet /
• lsmeans exertypetime / slicetime
  /differences in exertype for each time point/
• lsmeans exertypediettime / slicetimediet
• run
• quit

53
Worked Example from JL Text Book (Self Read)
54
Subject time 1 time 2 time 3 time 4
time 5 summary 1 y11 y12
y13 y14 y15 f(y11,
,y15) 2 y21 y22
y23 y24 y25 f(y21,
,y25)




n
yn1 yn2 yn3 yn4
yn5 f(yn1, ,yn5)
Summarizing with function over time removes
correlation Growth Curve
Approach
55
23 factorial design with temp, moisture and soil
type Each combination of factor level were
randomly assigned to two pots of soil Samples of
soil were taken in days 0, 7,14,30 and
60 Concentration of herbicide was measured for
each sample sphericity condition does not hold
56
(No Transcript)
57


58
xi day, n 5 yi log(concentration) f k
slope
sums
59
(No Transcript)
60
(No Transcript)
61
If you dont have a summary function, proc glm
can summarize with orthogonal polynomials over
time.
Linear orthogonal polynomial over time
62
Quadratic orthogonal polynomial over time