L4handout

# L4handout - PUBH 7430 Lecture 4 J Wolfson Division of...

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Unformatted text preview: PUBH 7430 Lecture 4 J. Wolfson Division of Biostatistics University of Minnesota School of Public Health September 15, 2010 A little bit of notation Notation - independent data With independent data on n units, for each unit j we have • (Scalar, one-dimensional) outcomes Yj • (Vector, p -dimensional) predictors xj Y1 Y2 . . . Yn x11 x21 x1 x12 · · · x1p x22 · · · x2p xn 1 x2 . . . xn2 · · · xnp xn Notation - correlated data With correlated data, units are grouped into K clusters. For each cluster i = 1, . . . , K we have • A vector of outcomes Y i corresponding to the units within cluster i : Yi = (Yi 1 , Yi 2 , . . . Yini ) • A matrix of covariates Xi where the rows correspond to the units within cluster i : x11 · · · x1p . . Xi = . xni 1 · · · xni p Note: We write ni for to allow for the number of units in each cluster to diﬀer; in many cases, ni is the same for all i Time-varying covariates x11 · · · x1p . . Xi = . xni 1 · · · xni p Covariates may be of two types: 1 Time-invariant. Eg. gender, height (for adults). Values are repeated down the appropriate column of Xi . 2 Time-varying. Eg. blood pressure, observation time. Values vary across rows of Xi Notation, cont’d. Putting this all together, the complete data can be represented by a “stacked” vector Y of length N = and “stacked” matrix X of dimension N × p : Y1 X1 . . . Y = . , X = . . YK XK Yij = Outcome for unit j from cluster i xijk = Covariate k for unit j from cluster i K i =1 ni Shorthand for summations and means Often, we will be summing within or across clusters. The “·” notation is a convenient shorthand: ni Yi · = Yij j =1 k ni x··k = xijk i =1 j =1 When we divide by the total number of terms summed up, the ¯ means we obtain are denoted by, eg., Yi · and x··k ¯ Expectation vector Expected value and variance extend to vectors in the following manner: Expected value The expectation of a vector Y is a vector whose entries are the expectations of the entries of Y: E (Yi ) = [E (Yi 1 ), . . . , E (Yini )] E (Y) = [E (Y1 ), . . . , E (YK )] Variance matrix Variance The variance (or variance matrix, or covariance matrix, or variance-covariance matrix...) of a vector Y is a matrix whose entries represent the covariance between the entries of Y: Var (Yi ) = Σ = Cov (Yi 1 , Yi 2 ) · · · Var (Yi 2 ) ··· . . . . . . Cov (Yini , Yi 1 ) ··· ··· Var (Yi 1 ) Cov (Yi 2 , Yi 1 ) . . . Cov (Yi 1 , Yini ) ··· . . . Var (Yini ) Covariance/correlation structures • Performance of most correlated-data models relies heavily on (implicit/explicit) speciﬁcation of covariance structure • For independent clusters, Var (Y ) = ΣY has a “block diagonal” structure: ΣY1 0 ··· 0 0 Σ Y2 · · · · · · ΣY = . . . . . . . . . . . . 0 · · · · · · ΣY K • For convenience/ease of estimation, often choose same simple structure for each ΣYi Exploratory data analysis Before you begin • Identify overall goal of study, scientiﬁc questions of interest • Identify: • Outcomes of interest • Predictors of interest (do they need to be included?) • Are outcomes/predictors time-invariant or time-varying? • Formulate statistical hypotheses, for example: • “β -carotene supplementation has no eﬀect on β -carotene levels over time” • “CD4 counts for HIV-infected patients are independent across time” • “Mouth cancer rates do not vary across diﬀerent regions of Scotland” • “Marriage/divorce do not predict weight gain in couples” Goals of Exploratory Data Analysis “Every good statistical analysis begins with an ’ocular test’, that is, a good look at the data” An exploratory data analysis should: • Show as much of the raw data as possible; summarize only when necessary • Highlight aggregate patterns within clusters • Highlight aggregate patterns across clusters • Identify unusual observations and clusters (outliers, etc.) Aspects of EDA 1 Trajectories and trends • Plots and smoothers 2 Correlation structures • The correlation coeﬃcient • The correlation matrix and autocorrelation • The variogram 3 Covariates • Residuals • Visualizing outcome-covariate relationships Trajectories and Trends: Plots and smoothers Spaghetti plot With data in long format, plot outcome versus time: 0.5 0.0 −0.5 log(FEV) 1.0 1.5 Log(FEV) vs. Age for 300 girls 6 8 10 12 14 Age 16 18 Spaghetti plot - three problems 0.5 −0.5 0.0 log(FEV) 1.0 1.5 Log(FEV) vs. Age for 300 girls 6 8 10 12 14 16 18 Age 1 2 3 Plots are crowded if number of observations is large Diﬀerences in individual variability can make trajectories hard to compare Systematic pattern in responses (eg. time/treatment eﬀect) can mask diﬀerences between individuals Detangling spaghetti Problem 1 Plots are crowded if number of observations is large Idea Restrict the number of individuals/clusters displayed by: • Taking a random sample • Making a grid of multiple plots 0 0 0 0 0 0 0 0 0 q 0 q 0 0 0 0 0 0 0 q q q 0 0 q 0 q 0 0 0 0 0 q 0 0 0 q 0 q 0 q 0 q q 0 0 0 q 0 0 0 q 0 q 0 q 0 0 q 0 0 0 q 0 q 0 0 0 Detangling spaghetti - grid of multiple plots 0 Within-person residuals Problem 2 Diﬀerences in individual variability can make trajectories hard to compare Idea “Standardize” each individual’s responses by subtracting (estimated) individual mean and dividing by (estimated) individual standard deviation: rij = where si2· = 1 ni −1 ni j =1 (yij yij − yi · ¯ si2· − yi · )2 ¯ Note: Within-person residual is undeﬁned for subjects with only one observation Within-person residuals 0 −1 −2 Individual Log(FEV) residual 1 2 Individual Log(FEV) residuals vs. Age for 50 girls 6 8 10 12 14 Age 16 18 Within-time point residuals Problem 3 Systematic pattern in responses (eg. time/treatment eﬀect) can mask diﬀerences between individuals Idea Standardize responses at time j by subtracting (estimated) mean and dividing by (estimated) standard deviation for all observations at that time: rij = yij − y·j ¯ s·2 j 1 where s·2 = #{obs (j )} i ∈obs (j ) (yij − y·j )2 , with the sum across ¯ j all i having an observation at time j . Within-time point residuals 1 0 −1 −2 −3 Within−time Log(FEV) residual 2 3 Within−time Log(FEV) residuals vs. Age for 50 girls 6 8 10 12 14 Age 16 18 Superimposed summaries Suppose we want to summarize the information contained in an outcome vs. time scatterplot. Simple approach Compute summary statistics (eg. mean, quantiles) at each time, and connect the dots: q qqq q q q q q q q qq qq q q q q q q q qq q q q q q q qqq q q qq q q q q q q q q q q q qq q q q q q q q qq qq q qqq qqqq qq q q q q qq qqq qq qqq qq q qq q qq q qqq q qqq qqqq qq q qqqq q q q q q q q qq qq q q qq q qq qq qq qq qq q qq q q qqq qq q q q q q q q q q q qqqqqqqqq qqq q qqqqq q qqq q qqq q q q q qq q qq q qqqqqq qq qq qqq q qqqqqqqqqqqq qqqqq qq qq qq qq q qq q qq qqq q q q q q qq q q q q q qqqqqqq qq q qqqqqq q q q q qq q q qq qqqqqqqqqq qqqqqq qqqqqqq qq q qq qq q qqq q q q q q qq qq qqqqqqqqqqqqqqqqq qqqqqqqq q q qq qq qqqqqqqqqqqqqqqqqqqqq qq q qq q q qqqqq qqqqq qq qq q qqqq q q q q q q q q qq qqqq q q q q q q q q qqq qq q q q q q qq qqqq q q qq qqq qq qqq qqq qq q qq q qq q q qq qq qq qq qq qq qqqq q q qq q q qqq qq q qq qqqq q q q q q q qq q q qq q q qq qqq qq qqqqqqqq qqqqqqqqq qqq qq qqq q qqqq q q q qq q q q qq q q q qqqq q q qqqq qqq qqqqq qqq q qq q qq q q q q qq qq qq qq qq q qq q q q q q qq q qq q q q qq q q q q qqq qq q q qq qq q q q qq q qqq qq q qq qqq qqq q qqqqqqqqqq qq qq qqq q q q q qqq q q q qqq qq qqq qqqqq qq q qqqq q q q q qqqqqqq qqq qqqq qqqq qq q q q qq q qq q q q q q q qq q q q q q q q q qq qqq q q qq qq q qq q q q q q q q qq qqq qq q q q q q q q q qq qq q qqqqqqqqqqq q q q qq q q q qq q q q q q q qq qq q qqqqqqqqq qqq q qq qq q qq q q q q q q q q q q qq q q q q qq q qq q q qqqq q qqqqqqqqq qq q q q q qq q q q q q qq qq q qq q q q q qq q qqqq q q qq qq qqq q q qq q q q qqqq q q q q qq q q q qq qq q q qq qq qq qq q q q qq q q qq q q q q q q q q qq q q qq q qqq q qqq q qq qq qqq q qqq q qq qq q qq q qq q q q qq q q q q q qq qqqq qq qqq qq qq q qq q q q q qq q qqq qq qqqqqqq qq qqq q q qqq q q qq qqqqqqqqqqq qqqqq q q q qq q qqq qq qq q q qqqqqqqqqqqqq qqqq q q q qq q qq q q q q q qqq q q q qq qqqqqq qq qqq q q q qq q q qqqq q q q q qqq q qq qq qq q q qq q qq q q q q q q q q q q q qqq q q q qq q q q q q qq q q q q q qqq qqqqq qqqqqqqq q qq q q q q qq qq q q qqqqq qq q qq q q q q q q qq q q q q q qq q qqq q q qqqqqqq qqq q q q q q q q qqq q qqqqq qq q q qq qqqq q qq q q q qqqq q q q q q qqq q qq q q q q qq q qq qqqq qqqqqqq q q q q q q q q q qqqq qq qq qq q q qqqqqqq qqqqqqqq q q qq qqqqqqqq qq q q q q q q qqqqqq q qq q q q qqq qq q q qq q qqqqqqqqqqq qqqqq q q q q qq q q qqqq qq q qqq q q q qq qq qqqqqq qq qq qq q q q q qq q q q qqqqqqq q qq q q q qqqqqqq q q q q qqq q q q q qqq qqqqq qqq qq q q q q q q q qq q qqqq qq qq q q qq q q q q q q qqqqq qq q qq qq q q q q qq q q q q q q qq q qqq qq q q q q q qq q q q qqqq qq q q q qqq q q q q q q q q q qq qq q q q q qq qq qq qqq qq qq q q q qq qq q q qq qq q q q q 8 10 12 14 Age q 0.5 0.0 Log(FEV) 1.0 0.75 0.5 0.25 −0.5 1.0 Log(FEV) 0.5 0.0 −0.5 Mean by age q 6 18 6 8 0 Quantiles by age q 16 1 q 1.5 1.5 q q qqq q q q q q q q q qq qq q q q q q qq q qq qq q q qq q q qq q q qq q q q qq q q q q qq qq q q q q q qq qq q q qq qqq q q q q qq q qq q q q qq q qq qq q qq q q 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q qqq q q qq qq qqqq q q qqq q q qqqq q q qq qq qq qqqqq q qq q q q q q q q qqqqq q q q q q q q q qqqqqqq q q qqqqq q qq q q q q qq q qqq qq q q q q qq q q qq q q q q q q q q qq q qqqqqqqq qqqqqqqqqqq q q qq qqq q qq qqqq qq q q q q q q q qq qq q qqqq qq qqq q qq q q q q qq q q q qq q qqq q qq q qq q qq q q qq q q q qqq qqqqq qqqqqqq q q q q q q q qq q q q q q q q q qqq q qq q qq qqq q q qq q q qqq qq q qqq qq q q q q q q q qq q q q q q qq q qq qq qq q q qq q qqqq q q q q q qq q qqq q q q q q qq q q q q qq q q q q q q qq qq q q qq qq q qq q q qq q q qq qq q q q q q 10 12 14 Age 16 18 20 Superimposed summaries But... • Summary statistics depend on how observations are grouped by time (eg. mean for 6 year olds, or 6-9 year olds?) • May be sensitive to outliers (esp. means) Scatterplot smoothing Goal Estimate the underlying mean response curve µ in the model Yi = µ(ti ) + i In the FEV example: log(FEVi ) = µ(Agei ) + i Kernel smoothing Basic concept of kernel smoothing • To get an estimate of the mean response at t , average responses of observations near t . • Repeat for every value of t to get mean response curve Questions • What is “near”? • How do we “average”? Smoothing windows What is “near”? • Use points within a certain distance of t • This distance (called a bandwidth, often denoted by h) deﬁnes a horizontal “window” surrounding t 1.5 q q q q q qq q q q qq qq qq q qq q q q q q q q q qq q q q q qqqqq q qqqq qq q qqq q q qqq q qq q q qqq q q qq q q qq qq q q q qq qqqq q qqq qq qq q q q q qq q qqqq qq q qq q q q qq q q q qq q q q qq q q q q q q qq q q q q qq qq qqqqq qqqq qq q qqq q qq qq q qq q qqq q qqq q q q q qqqq q q qqqqqq qqqqqqqq q q qq qqqq qqqq qqqq qqqqqqqqq q q qq q qq qqq q q qqqq qq q qqqqq qq qq qqq qq q q qq q qq q q qq q qqqq qq qq q q q qqqq qq q qq qq q qq q q q q qq q qq qq qqq q q q q q qq q q q q qq q q qq qq q qqqqqq q qqqq qq q q qq qq q qq qqq q qq qq q q q qq qq q q qq qq q q q q q qq q q q q qqq q qqqq qqqqqqqqqqqqqqqqqqqqqqq q qqq qqqqqq qqqq qqq q qq q q q qq q q q qq q q q q q qq q q qq q q q q q q q qq q q qq q q qq q q qq qq q q q q q q q qqq q q q qq qqqq qqqq qqq qq q q qqq qq qq qq q q qqqqqqqq q qqqqqqq qqqqqq qq qq qq qq q q q q qqqqqqqq qqqq q q q q q q q qq q qq q q qq q q q q q q q q q q q qq q q q q q qqqq qqq q qq qqqqqq qqqqqqqq qq qq qq q q qqq qqqq qq q qqq q qq qq qqq q qq q qq q qq q q qq q q qq q qqq q q q q q qqqq q q q q q q qq q qqq q qq q qq q q q qqqqq q q q qq q qq q q q q q q q qqq q qqq qqqqqqq q qqqqqq qqqq q qq q q qq q qqq qq qqqq qqq qq q q qq q q q qq q q qq q q q q qq q qqqq q qq q q q q qq qq qq q q qq q q qq q q q q q qqq q qq q q q q q qqq qqq qqqqq q q qq q q qqq q qq q qqqq q q q qqq qqqq qqqqqqqqqqqq q qqq q q q q q q q qq qq q qq q q q qq qq q q q q q q q q qq q q q q qq q q q qq q qq q q q q qq qq qq qqqq q q q q q q q q qq q q qq q q q q qq q qq q q q qq qq q qq qq q q qq qq q q qq q q qq qqq q qq q qqqq q q q qq q qqq qq qq q qq qq qqq q q q q qqq q q q q q q q qqqq qq q q q qq q qqqq q qq q qqq q q qq q q q q q qqq qq q q qqqq q q q q q q q qq q q qq q q q q q q q qq q qq q q q q qq qq qq qqqqqqqq qqq q qqqqqqq qqq q qqq qq q q q q q qq qq q q q q q qq q q qqq q qq q q qqqq q qqq q q q q q q q q qq q qq q q q qq qqq q qq qq q q q q q qq q q qq q q q qqq qq q qq qqqq qqq q qqq qqqqqq q qq q q qqq q q q q qq q qq qq q q qq qqqq qqq qqqq q qqqqqq qqqqqq qqqqqqq q q qq q q q q qq qq q qq q qqq qq q q qqq q qq q q qq qq q q q q qq q q qqqq qqqqqqqq q q q qq qqqqq qqq q q q q q q q q q qq q q qq q q q q qq qqqq q q qq q q q q q qqq qqqq qqqqqq qqqq qq qqqq q q q q qq qqqqqq qq q qq q q qq q q q q q qq q qqq q q q qq q q qqq qq q q q q qq q qq q q q q q q qqq q q q q q qq q q qq q qqqqqqq qqqqqq qqqqq q q qq qqq q q q qq q q qq q q q q qqqqqqq qqq qq q q qq qqq q q qq qq qq q q q qq q q qqq qq q qqqq q q qqq qq qqqqq qqqqq qq q q q q q q q q q q q q q q qq qqqqq q qq q q qqq qq q q q qq qq qqq q q qq q q q q q q qqq qqq q qqq q q qq qq q q q q qqq q qq q q q q qq q q qq qq q q q q q q q qq q qq q qqq qqqqq q q q q q q qq q q q qq q q qq q qq qq q qq 0.5 0.0 q −0.5 Log(FEV) 1.0 q Bandwidth = 2 years q 6 8 10 12 14 Age 16 18 Smoothing weights How do we “average”? • Obvious way is to take mean of all points within a window • More generally, can assign weights to points according to how far they are from t , i.e. points on the “edges” of the window count less than points near the center • Mathematically, deﬁne µ(t ) = ˆ n i =1 K [(t − ti )/h ]yi n i =1 K [(t − ti )/h ] • K is known (for unimportant mathematical reasons) as a kernel function Kernel functions Many diﬀerent kernel functions are possible. Popular options include: • “Boxcar” kernel: straight average of points within a window • Gaussian kernel: K (u ) = exp (−0.5u 2 ), weight decays exponentially with distance from t Boxcar kernel Gaussian kernel 8 10 12 Age 14 16 18 1.5 1.0 Log(FEV) 0.5 1.0 Log(FEV) 0.5 0.0 6 q q q q qq q qq qq q q q q q q q q q qqq q qq qq q q q qq q q q q q q qq q qq q q q q qq qqqqqqqqqq qq qq q qq q q q q qq q q q q qqqq qq qq q q q qq q qqq q q qqqqqqqqq q qq q q qq q q q qq q qq q q q q qq q q q qq q q q q qqq q qqq q q q qq q qq qqq qqqqq q qq qqqqqqq qqq q q q q q qq q q qqq q q q q q qqqqqqqqqq qqqqq qq q q qq q q q q q qq q q q q q q qq q qq qqqqqqqqqq qqqqqqq q q q qqq qqq qqqqqqq qqqqqqqqq qq q q qq q q qqq qqq q qq q q q qqq qq qqqq qq qq q q q qq q q qq qqqq q q q qqq q qqq q qqqqqq qqqqqqqqqqqqqqqqqqq qqq qq qqqqqqqqqq qqq qqqq q q q q qqqq q qqqq qq qq qqq q q qq q q q q qqq qq qq qqqq q q q q qq q qq qqq qq qqq qqq qqqqqq q q q qq qq q q q qq q q q q q qqqq q q qq q qq qq q qq q q q qq q qq qq qq qqq qqqq qqqqqq qq qqqqq q qq qq qqqq q q qqq qqq q q q q qq q qqqq q q q q q q q q q q qqqq q q q q q q qq qq q qqqqqqqqq qqqq q qq qq q q q q q q q q qq q q qq q q qqq q q qqqq q q q qq q q q qqqqqqq qqq qqq qq q q qqqqq q q qqqq qq q q qq qq q q qq q q qqq q q q qqq q q q q q q q q q q qqqqqqqq q q q qq qq q qq q qq q q qqq q qqqqqqqqqqqqq q qq q q q q qqq q q qq qq q q q q q q q q q q q q q qq q qq q qqqqq q qqq q qq qq q qqqqqq q q qq q qq qq qqqqqqqqqqqqqq q qq q qqq q q q q q q q q qqqqqqqq qq q q q qq q qqqqq q q q qqq q q qqq q qq qq q q q q qqq q qq qqqq q q qq qqq q qq q qq q q qq qq qqqqqqqq q q q qq qqqqq qqqqq q q qqqqq qq q qq q q qq q qq qq q q q q qq q q q q qq q q q qq q q q qq q qqq q q q q q q q qq qq qqqq qqq q qq q q q q q qqq q q qqqqqqq q qqq qqqqq q q q qqqqqqqqqqq qqqqqqqq q q qqqqqqqq q qq q q q q qq qq qq qqqqq qq q qqqqqq q q qqqqq q q q qqqqq q qq q q q q q q q qq q q q q q q qq q q q qqqqqqqq qq q qq q q q qq qqq q qq q qqqqqqqqqqq qqqqq qqq q q qqqqqqq q qq q qqqq q qq qq qqqqqqqqqqqqqqqq q q qqqqq qqqqqq q q q q q q qq qq q q q qq q q q qq qqqqqqq qq qq q q qq q q q qq q q qq q q q q q q q q q q q qqq qq q qq q q q qqqqqq q q qqq q q q qqqqq q qq q q q qq qq q q qqqqqq qq q q q q q q qqq q qq q q qqq qqqqqqq q q qq q q q qqq qq q q q qqqqq qqqq qq qqq qqq q qqqqqqq qq q qqqqqqq qq q q q q qqqqq qqq qqq q q q qq q q qqqqqqqq qqq qq qq q q qq q q q qq q qqqqqq qq qqq qq q qq q q q qqqq q q q q q qqq q q qq q qqqqq q qq qq q q q q qq q qqq q q qq q qq q q qq q qqq q q q qq qq q qq q q q qqqqqq qq q q qqqq q q qq q q qq q qqqqq q q qqq q q q q q qq qq qq qq qq q q q q q q q qqq qqq q q q qq q q qq qqq q q qq qq q q qq q qq qqq q q q qq qqq q qq q qq q q qq q q qq q q 0.0 1.5 q q q q qq q qq qq q q q q q q q q q q qqq q qq qq q q q qq q q q q q q qq q qq q q q q q qq qqqqqqqqqq qq qq q qq q q q q qq q q q q qqqq qq qq q q q qq q qqq q q qqqqqqqqq q qq q q qq q q q qq q qq q q q q qq q q q qq q q q q qqq q qqq q q q qq q qqq q q qq q q q q qq qqqqq q qq q qqqqqq qqq q q q q q qq q q q q q q qqqqqqqqqq qqqqq qq qq q q q qq q q q q q q qq qqqqq qqqqq qqqqqqq q q q qqq qqq qq qqqqq qqqqqqqqq qq q q qq q q qqq qqq qq q q q qq q q q q qqq qq qqqq qq qq q q q qq q q qq qqqq q q q qqq q qqq q qqqqqq qqqqqqqqqqqqqqqqqqq qqq qq qqqqqqqqqq qqq qqqq q q q q qqqq q qqqq qq qq qqq qqq q q q qqq qq qq qqqq q q q q qq q qq qqq qq qqq qqq qqqqqq q q q qq qq q q q qq q q q q q qqqq q q qq q qq qq q qq q q q qq q qq qq qq qqq qqqq qqqqqq qq qqqqq q qq qq qqqq q q qqq qqq q q q q qq q qqqq q q q q q q q q q q qqqq q q q q q q qq qq q qqqqqqqqq qqqq q qq qq q q q q q q q q qq q q qq q q qqq q q qqqq q q q qq q q q qqqqqqq qqq qqq qq q q qqqqq q q qqqq qq q q qq qq q q qq q q qqq q q q qqq q q q q q q q q q q qqqqqqqq q q q qq qq q qq q qq q q qqq q qqqqqqqqqqqqq q qq q q q q qqq q q qq qq q q q q q q q q q q q q q qq q qq q qqqqq q qqq q qq qq q qqqqqq q q qq q qq qq qqqqqqqqqqqqqq q qq q qqq q q q q q q q q qqqqqqqq qq q q q qq q qqqqq q q q qqq q q qqq q qq qq q q q q qqq q qq qqqq q q qq qqq q qq q qq q q qq qq qqqqqqqq q q q qq qqqqq qqqqq q q qqqqq qq q qq q q qq q qq qq q q q q qq q q q q qq q q q qq q q q qq q qqq q q q q q q q qq qq qqqq qqq q qq q q q q q qqq q q qqqqqqq q qqq qqqqq q q q qqqqqqqqqqq qqqqqqqq q q qqqqqqqq q qq q q q q qq qq qq qqqqq qq q qqqqqq q q qqqqq q q q qqqqq q qq q q q q q q q qq q q q q q q qq q q q qqqqqqqq qq q qq q q q qq qqq q qq q qqqqqqqqqqq qqqqq qqq q q qqqqqqq q qq q qqqq q qq qq qqqqqqqqqqqqqqqq q q qqqqq qqqqqq q q q q q q qq qq q q q qq q q q qq qqqqqqq qq qq q q qq q q q qq q q qq q q q q q q q q q q q qqq qq q qq q q q qqqqqq q q qqq q q q qqqqq q qq q q q qq qq q q qqqqqq qq q q q q q q qqq q qq q q qqq qqqqqqq q q qq q q q qqq qq q q q qqqqq qqqq qq qqq qqq q qqqqqqq qq q qqqqqqq qq q q q q qqqqq qqq qqq q q q qq q q qqqqqqqq qqq qq qq q q qq q q q qq q qqqqqq qq qqq qq q qq q q q qqqq q q q q q qqq q q qq q qqqqq q qq qq q q q q qq q qqq q q qq q qq q q qq q qqq q q q qq qq q qq q q q qqqqqq qq q q qqqq q q qq q q qq q qqqqq q q qqq q q q q q qq qq qq qq qq q q q q q q q qqq qqq q q q qq q q qq qqq q q qq qq q q qq q qq qqq q q q qq qqq q qq q qq q q qq q q qq 6 8 10 12 Age 14 16 18 Bandwidth • Wider windows (larger bandwidths) give smoother estimates • Narrower windows (smaller bandwidths) give bumpier estimates Large bandwidth Small bandwidth q q q qq q qq qq q q q q q q q q q qqq q qq qq q q q qq q q q q q q q q qq q q q q q q qq qqqqqqqqqq qq q q q qq q q q q q qq q q q qqqq q qq q q q qq qq q qqq q q qqqqqqqqq q qq q qq q q q qqqq qq q q q q q q q qqq q qq q qq q qq q qq q q qqq qqq q qq q qqq q q q qqq q q qq qq q q qqqq q q q qqqq q qqqqqq q qq qqqq qq q qqqqqq qqq q q q qq qq q q q q q q q qq q qqqqq qqqq qqq q q qq qqqq qq qqqqq qqqqqqqqq qqqq q q q q qqq qqqqqqqqqq q qq qqq qq q qq qqqqq qqqqq qqqqq qq q qq q qqqq qq qq q qqqqq qqq q qqq q q q qq q q qqqqqq qqq qqqqqqqqqqqqq q qqq qq qq q qqqqq qq q q q q q qq qq qqqq q q q qq q qq q q q q q qq q qq q q qq q qq qqq qqqqq qqq qqqqqq q q q qq qq qq q qq q q q q qqqq q q qq q qq qq q qq q q q qq q qq qq qq qqq qqqq qqqqqq qq qqqqq q qq qq qqqq q qq qq qqq q q q q qq q qqqq q qqq q q q q q q q qqqq qq q q q q q qq qq q qqqqqqqqq qqqq q qq qq q q q q q q q q qq q q q qq q qqqqqq qqqqqq q q qq q q q qqqqqqq qqq qqq qq q q qqqqq q q qqqq qq q q qq qq qq q q q q q qqq q q q qqq q q q q q q q q q q qqqqqqqq q q q qq qq q qq q qq q q qqq q qqqqqqqqqqqqq q qq q q q q qqq q q qq qq q q q q q q q qq q q q q qq q q qqqq q qqq qq q q qq q q qqqqqq q q q qq q qq qq qqqqqqqqqqqqqq q qq q qqq q q q q q q q q qqq qq qqq qq qqqqqqqq qq q q qq qq q qq qq q qqqq qq qq q q q qq qq q qq q q q qq q q q qq q qq q qq q q qq qq qqqq q qq q q qq qqqqq qqq qq q q q q qq q qqq qq qqq qqq q q q qq q q q q qqq q q qq q q q qq q qqq q q q q q q q qq qq qqqq qqq q qq qq q q q qq qqq q q q q qqqqqqqq qqq qqqqq q q qq q q qq q q qqqqqqqqqqq qqqqqqqq q q qqqqqqqq q qq q q qq qq q q qq q qqqqqq q q qqqqq q q q qqqqq q qq q q q q q q q qq q q q q q q qq qqqqqqqq qq q qq q qq q q qqq qqq q qq q q qqqqqqqqqqq qqqqq qqq q qqqqqqqqq q qq q q q q qq q qq q qq qq q qqqqqqqqqqq q q q qq q qqqqqqqqq qqqqqq qq q q q qq qqqqqqqq q q q q q q q qqq q q qq q q q q q q q q qqqqqqqqq qqq q q qq q q q q q qq q q qqqq qqq q qq q q qqqq q q q q q qq q q qqqqqq qq q q q q qqq q qq q qqq qqqqqqq q q qq q q q qqq q q qqqqq qqqq qq qqq qqq q qqqqqqq qq q qqqqq qq q q q qq q q q qq q qqqqq qq q qqq q q qq q q q qqqqqqqq qqq qq qq q q qq q qqq q qqqqqq q q qqq qq q qq q q q qqqq q q q q q qqq q q qqq qqqqq q qq qq q q q qq q q q qq q q q qq q qq q q qq q qqq q q q qq qq q qq q q q qqqqqq q q q q qq q q qq q q q qqq q q qq qq q qqq qq q qq q qq qq q q qq qq q q q qq q q q q q qqq qqq q qq q qq qqq q q qq qq q q qq q qq qqq q q q qq qqq q qq q qq q q qq q q qq q 1.5 1.5 q q q q qq q qq qq q q q q q q q q q qqq q qq qq q q q qq q q q q q q q q qq q q q q q q qq qqqqqqqqqq qq q q q qq q q q q q qq q q q qqqq q qq q q q qq qq q qqq q q qqqqqqqqq q qq q qq q q q qqqq qq q q q q q q q qqq q qq q qq q qq q qq q q qqq qqq q qq q qqq q q q qqq q q qq qq q q qqqq q q q qqqq q qqqqqq q qq qqqq qq q qqqqqq qqq q q q qq qq q q q q q q q qq q qqqqq qqqq qqq q q qq qqqq qq qqqqq qqqqqqqqq qqqq q q q q qqq qqqqqqqqqq q qq qqq qq q qq qqqqq qqqqq qqqqq qq q qq q qqqq qq qq q qqqqq qqq q qqq q q q qq q q qqqqqq qqq qqqqqqqqqqqqq q qqq qq qq q qqqqq qq q q q q q qq qq qqqq q q q qq q qq q q q q q qq q qq q q qq q qq qqq qqqqq qqq qqqqqq q q q qq qq qq q qq q q q q qqqq q q qq q qq qq q qq q q q qq q qq qq qq qqq qqqq qqqqqq qq qqqqq q qq qq qqqq q qq qq qqq q q q q qq q qqqq q qqq q q q q q q q qqqq qq q q q q q qq qq q qqqqqqqqq qqqq q qq qq q q q q q q q q qq q q q qq q qqqqqq qqqqqq q q qq q q q qqqqqqq qqq qqq qq q q qqqqq q q qqqq qq q q qq qq qq q q q q q qqq q q q qqq q q q q q q q q q q qqqqqqqq q q q qq qq q qq q qq q q qqq q qqqqqqqqqqqqq q qq q q q q qqq q q qq qq q q q q q q q qq q q q q qq q q qqqq q qqq qq q q qq q q qqqqqq q q q qq q qq qq qqqqqqqqqqqqqq q qq q qqq q q q q q q q q qqq qq qqq qq qqqqqqqq qq q q qq qq q qq qq q qqqq qq qq q q q qq qq q qq q q q qq q q q qq q qq q qq q q qq qq qqqq q qq q q qq qqqqq qqq qq q q q q qq q qqq qq qqq qqq q q q qq q q q q qqq q q qq q q q qq q qqq q q q q q q q qq qq qqqq qqq q qq qq q q q qq qqq q q q q qqqqqqqq qqq qqqqq q q qq q q qq q q qqqqqqqqqqq qqqqqqqq q q qqqqqqqq q qq q q qq qq q q qq q qqqqqq q q qqqqq q q q qqqqq q qq q q q q q q q qq q q q q q q qq qqqqqqqq qq q qq q qq q q qqq qqq q qq q q qqqqqqqqqqq qqqqq qqq q qqqqqqqqq q qq q q q q qq q qq q qq qq q qqqqqqqqqqq q q q qq q qqqqqqqqq qqqqqq qq q q q qq qqqqqqqq q q q q q q q qqq q q qq q q q q q q q q qqqqqqqqq qqq q q qq q q q q q qq q q qqqq qqq q qq q q qqqq q q q q q qq q q qqqqqq qq q q q q qqq q qq q qqq qqqqqqq q q qq q q q qqq q q qqqqq qqqq qq qqq qqq q qqqqqqq qq q qqqqq qq q q q qq q q q qq q qqqqq qq q qqq q q qq q q q qqqqqqqq qqq qq qq q q qq q qqq q qqqqqq q q qqq qq q qq q q q qqqq q q q q q qqq q q qqq qqqqq q qq qq q q q qq q q q qq q q q qq q qq q q qq q qqq q q q qq qq q qq q q q qqqqqq q q q q qq q q qq q q q qqq q q qq qq q qqq qq q qq q qq qq q q qq qq q q q qq q q q q q qqq qqq q qq q qq qqq q q qq qq q q qq q qq qqq q q q qq qqq q qq q qq q q qq q q qq 6 8 10 12 Age 14 16 18 0.5 Log(FEV) 1.0 q 0.0 0.0 0.5 Log(FEV) 1.0 q 6 8 10 12 Age 14 16 18 ...
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