Azevedo, Cecília

Correia-Neves, M

Horta, Ana Maria Lacerda Morgado Fernandes Carvalho Aboim

Silva, Rita Catarina Assunção Ribeiro

Universidade do Minho: RepositoriUM

On the determination coefficient and global adjustment test of goodness of fit

On the Determination Coefficient and GlobalAdjustment Test of Goodness of FitHIV infection :: CD4+ T cells dynamicsCecilia Azevedo; Rita Rb-Silva; Ana Horta; Margarida Correia-NevesCentre of Mathematics, School of Sciences & Life and Health Sciences Research Institute, Schoolof Medicine.University of MinhoTable of contents1. Introduction2. Motivation and data set3. Statistical methodology4. Results5. Discussion1IntroductionCD4+ T CellsT cells are regulators of the immune system. CD4+ T cells are alsoknown as helper T cells because they act on other cells of the immunesystem to promote various aspects of the immune response. They have acrucial role in the bodys immune system.Figure 1: Human CD4+ T Cell. Image courtesy of NIAID2HIV - Human immunodeficiency virusWhen HIV virus enters in the bloodstream of a person it mainly attacksCD4+ T cells, thought its CD4 protein (located in the border of the cell,Figure 2).Figure 2: Membrane fusion3HIV - Human immunodeficiency virusThe HIV virus is a retrovirus as its genome is a single-stranded RNA andinclude in their replication cycle DNA intermediates (Figure 3).Figure 3: Viral RNA in the host cell cytoplasm4The virus uses its own reverse transcriptase enzyme to produce DNAfrom its RNA genome (Figure 4).Figure 4: Viral DNA5That DNA is incorporated into the host cell genome who treats the viralDNA as part of its own genome, translating and transcribing the viralgenes and producing the proteins required to assemble new copies of thevirus (Figure 5).Figure 5: New copies of HIV6Number of CD4+ T cellsThe HIV replication results in massive depletion of CD4+ T cells in thehuman blood. After this acute phase, several immune responses aregenerated by the host and viremia declines until a lower state level.However these responses fail to eliminate the virus leading to a chronicinfection in most individuals during an asymptomatic period which cango on several years. During the chronic phase of the infection, bloodCD4+ T cell count declines slowly (see Figure 6).Figure 6: Schematic diagram of the course of HIV-1 infection7Motivation and data set::The problem::AntiRetroviral Treatment (ART) dramatically suppresses viral replicationand reduces the plasma HIV-1 viral load to below the limits of detectionof the most sensitive clinical assays (<50 RNA copies/mL).ART causes a progressive reconstitution of the immune system.That must imply an increase in circulating CD4+ T-lymphocytes.The mathematical problem includesgrowth significance :: test if T-lymphocytes growth during ART isstatistically significant;growth magnitude :: quantify globally the increase in circulating CD4+T-lymphocytes during ART.8:: The data ::We have CD4+ T cells counts of 50 patients (10 female) with HIV-1 andin the same ART program. The counts were taken at baseline (BL) andat 1, 3, 6, 9, 12, 16, 20, 24, 28, 32, 36, 42, 48, 54, and 60 months duringART. In this problem we are considering data for male and femaletogether because there is no statistical evidence that this two groups aredifferent concerning to CD4+ T cells counts at the different time points,age, or even body mass index (BMI at BL).Table 1: Characteristics of the patients in the beginning of ARTMin. 1st Qu. Median Mean 3rd Qu. Max.CD4 BL 9.0 116.0 262.5 247.1 320.5 1033.0 186.44Age 22.00 32.75 41.00 40.55 46.25 67.00 10.24BMI 16.23 20.75 23.05 23.45 25.82 33.61 3.4879Statistical methodologyThere are an enormous variability in CD4 count data. This variability iswithin subjects and between subjects (as usual in this type of humandata). In Figure 7, we have the data for the different subjects in separatepanels.Time pointsCD4 count0500100015000 20 40 60ll llllll lllllAH039lllll lllllll lAH0150 20 40 60ll lllllll llllllAH073lllAH0020 20 40 60llllAH029l llll llAH0010 20 40 60llll lllllllllAH048lllll lllllll lllAH079l lllllllllAH037ll lllllllllllAH090llllllllllAH013lllllllll lllllAH089lllllllll lllllAH006l lllll llllllAH035ll lllllll llllllAH088050010001500lllll llllllll llAH080050010001500l llAH072l lllllllllllllAH052lllllllllllll lAH066ll lllllll llllAH025llll llllllll lllAH078ll llllllll lllllAH058llll llllll lllAH092l lllllll lllllAH036llllllllllllllAH062lllllll llllllAH004ll lllll llllllAH022llllllllll lllllAH075ll llllllllllllAH042ll llllll lllllAH040llll lllllllllAH047050010001500llllllllllllll lAH083050010001500lll lllllAH009lll llllllllllllAH061lll llllllll llllAH076llllllllAH054llllll llllll lAH049llllll lllllll lAH096ll llllll lllllAH007ll llllll lllllAH041lllllll llllllAH021lllllllllll lAH060llllllll lllllAH023llll lllllllll llAH093lllllllllll lllAH027lll llllllllllAH045ll llllllAH059050010001500llll lllllllllllAH046050010001500llllllllll llllAH0260 20 40 60lllll lllllll llllAH077Figure 7: Scatter diagram10A Linear ModelGlobally CD4+ counts grow over time, see Figure 8.350400450500550600650Mean of CD4 T cells number with timetimecd4_countlll lllllll lllllll0 1 2 3 6 9 12 20 28 36 48 60n=41n=43n=39n=43n=34n=42n=37n=43n=39n=44n=38n=41n=35n=43n=41n=39n=40Figure 8: Global mean growth of number of T cells with time11ResultsCoefficient of DeterminationIf we use all data (all counts available for the 50 patients), we obtain theequation y = 455.887 + 1.628x for least-squares regression line and thislinear model has a coefficient of determination very low, R2=0.012, somore than 98% of the total variability on data is not explained by thismodel, see Figure 9. The model explains nothing!!!llllllllllllllllllll llllllllllll llllllllllllllll llllllllllllll llllllllllll llllllllllllllllllllllll llllllllllllll lllllllllllllllllll lllllllllllllllllllll llll lllllllllllllllllllllll llll lllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllll lllll ll llllllllllllllll ll llllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllll lllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllll llllllllllllllllllllll lllllllllll llllllllllllllllllllllllllllllll0 10 20 30 40 50 60050010001500timeCD4+ countFigure 9: Scatter Plot with Regression Line12Utility of the modelThe model presented is an useless model?Global Adjustment Test based on β (the slope parameter)H0 : β = 0 (useless model) vs H1 : β 6= 0.The test statistic isT =β̂σ̂β̂=β̂√MSEs2x√n − 2 ∼ tn−1and the p-value = 2× (1− Φ(n − 1, |Tobs|)), where Φ is the df of tn−2and |Tobs| is the absolute value of the observed test statistic T . In ourcase we have the estimative of the parameter, β̂ = 1.63 and a lowstandard error associated σ̂β̂ = 0.572 which implies a statistic valueTobs=2.842.In this case we have 680 degrees of freedom (50×17-168-2), so the pvalue is 0.0046, and we reject the useless of the model.13DiscussionConclusionsThe low p-value of the Global Test of Adjustmentis a consequence of the large number of observations.For medicine is a realistic model. There are a global (constant) meangrowth of 20 T cells per year of ART. As far as we know is consensualthat there are a global growth of this kind of cells during ART but therewas not a quantification of the growth.Here we quantify the global growth but this model should not be used tomake predictions.This result allows answering a part of the question:if there is growth, it is a slow one. It should not be forgottenthat only the positive balance is being considered. It has to betaken into account growth that prevents decay.14to be continued . . .THANK YOU VERY MUCH FOR YOUR ATTENTION !!!!15

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On the determination coefficient and global adjustment test of goodness of fit

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Universidade do Minho: RepositoriUM