Predicting Envelope Leakage 

in Attached Dwellings

O. Faakye, L. Arena, D. Griffiths

Consortium for Advanced Residential Buildings

July 2013
NOTICE

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iii

Predicting Envelope Leakage in Attached Dwellings

Prepared for:

The National Renewable Energy Laboratory

On behalf of the U.S. Department of Energy’s Building America Program

Office of Energy Efficiency and Renewable Energy

15013 Denver West Parkway

Golden, CO 80401

NREL Contract No. DE-AC36-08GO28308

Prepared by:

O. Faakye, L. Arena, D. Griffiths

Steven Winter Associates, Inc.

of the

Consortium for Advanced Residential Buildings (CARB)

61 Washington Street

Norwalk, CT 06854

NREL Technical Monitor: Cheryn Metzger

Prepared under Subcontract No. KNDJ-0-40342-03

July 2013
iv

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v

Contents

List of Figures ............................................................................................................................................ vi

List of Tables .............................................................................................................................................. vi

Definitions.................................................................................................................................................. vii

Acknowledgments ................................................................................................................................... viii

Executive Summary................................................................................................................................... ix

1 Introduction and Background ............................................................................................................. 1

2 Research Objectives ............................................................................................................................ 3

3 Technical Approach ............................................................................................................................. 4

4 Model Development Methodology ...................................................................................................... 5

4.1 Multivariate Linear Regression Analysis.............................................................................5

4.2 Predictor Variables Considered ...........................................................................................8

4.3 Major Predictive Variables..................................................................................................9

4.4 Multivariable Linear Model Results..................................................................................11

4.4.1 Random Forest Approach ......................................................................................16

4.4.2 Area Weighted Approach ......................................................................................17

5 Comparing Models ............................................................................................................................. 19

6 Preliminary Test Case ........................................................................................................................ 21

7 Discussion........................................................................................................................................... 23

8 Conclusion .......................................................................................................................................... 24

References................................................................................................................................................. 25

Appendix A: Predictor Variables Considered ........................................................................................ 26

Appendix B: Random Forest Output....................................................................................................... 29
vi

List of Figures

Figure 1. Process for creating model........................................................................................................ 6

Figure 2. Illustration of typical building specification............................................................................. 8

Figure 3. Plot of MVLR model fitted against measure RFS .................................................................... 13

Figure 4. Plots validating linear model ................................................................................................... 14

Figure 5. Plot of Rstudent residuals against predictors....................................................................... 15

Figure 6. RF model results....................................................................................................................... 16

Figure 7. RESATSA model results .......................................................................................................... 18

Figure 8. Floor plan of KCHA apartment units....................................................................................... 21

Unless otherwise noted, all figures were created by CARB.

List of Tables

Table 1. Summary of Building Used.......................................................................................................... 4

Table 2. Coding Scheme for Categorical Variables................................................................................. 9

Table 3. Initial MVLR Model Results.......................................................................................................... 9

Table 4. MVLR Model Results .................................................................................................................. 11

Table 5. MVLR Model Equations.............................................................................................................. 12

Table 6. Summary of Regression Accuracy........................................................................................... 19

Table 7. MVLR Model Predictions Versus Sample Data........................................................................ 22

Unless otherwise noted, all tables were created by CARB.
vii

Definitions

CARB Consortium for Advanced Residential Buildings

CFM50 Cubic feet per minute of airflow needed to create a change in 

building pressure of 50 Pascal

ESA Exposed surface area of building envelope

FG Fully guarded blower door value ( CFM50)

LTO Leakage to outside (CFM50)

MVLR Multivariable linear regression

PCC Partial correlation coefficient

RCWTSA Ratio of common wall to total surface area

RESATSA Ratio of exposed surface area to total surface area

RF Random Forest

RFS Ratio between fully guarded and solo blower door values

RWATESA Ratio of window area to total exposed surface area

SO Solo blower door value (CFM50)

TSA Total surface area of building envelope
viii

Acknowledgments

CARB would like to thank those who have enthusiastically supported this effort. A special 

thanks for the input of project advisors Michael Lubliner, Terry Brennan, Gary Nelson, Collin 

Olson, and Iain Walker. We are also grateful to Anthony Cox and Chase Counts of the New 

River Center for Energy Research and Training for sharing a treasure trove of data. 
ix

Executive Summary

The most common method for measuring air leakage is to use a single blower door to pressurize 

and/or depressurize the test unit. In detached housing, the test unit is the entire home and the 

single blower door measures air leakage to the outside. In attached housing, this “single unit,” 

“total,” or “solo” test method measures both the air leakage between adjacent units through 

common surfaces as well air infiltration or exfiltration across the exterior, noncommon surfaces

of the enclosure. Measuring and minimizing this total leakage are recommended to avoid indoor 

air quality issues between units, reduce energy losses to the outside, reduce pressure differentials 

between units, and control stack effect. However, two significant limitations of the total leakage 

measurement in attached housing are:

• For retrofit work, if total leakage is assumed to be all to the outside, the energy benefits 

of air sealing can be significantly overpredicted. 

• For new construction, the total leakage values may result in failing to meet an energybased house tightness program criterion.

A practical method needs to be developed to quantify the fraction of total leakage for an attached 

dwelling that is to the outside. The consensus of a Building America Expert Meeting held in 

March 2012 confirmed the need for research to develop such a method.

Using blower door test data available from four multifamily projects, the framework for a simple 

algorithm based upon a solo blower door test result and a few basic dwelling unit characteristics 

has been outlined. While the subject dataset is very limited, preliminary analyses suggest that 

statistically significant predictors are present and can support the development of an algorithm. 

The key next step is to collect additional data for analyses and algorithm development so they 

may be more broadly applied with confidence.

The scope of this research is to investigate an approach for developing a viable simplified 

algorithm that can be used by contractors to assess energy efficiency program qualifications

and/or compliance based upon solo test results. This report describes the work that has been done 

thus far and is intended for the building science research community that is familiar with blower 

door test protocols. The algorithm would not replace the more rigorous, and more accurate, 

guarded blower door method that is appropriate for building science research. Also, this research 

effort does not intend to suggest appropriate targets for maximum air leakage values.
1

1 Introduction and Background

Guarded blower door testing, or fully guarded (FG) testing, is a pressurization method that is 

often recommended for measuring air leakage to the outside. This method utilizes multiple 

blower doors to pressurize or depressurize adjacent spaces to the same level as the unit being 

tested; maintaining a neutral pressure across common walls, ceilings, and floors acts as a “guard” 

against air exchange between units. The measured air leakage in the test unit is air leakage to the 

outside. While preferred for assessing energy impact, this method is often not implemented

because the equipment and labor requirements can be daunting. In retrofit situations where 

adjacent units may be occupied, simultaneous access can be logistically difficult. These 

challenges have been noted by other Building America researchers as well (Lyons 2013;

Neuhauser et al. 2012; Ueno et al. 2012; Wytrykowska et al. 2012).

Whole-building testing is analogous to guarded blower door testing because both methods

attempt to measure energy-related exterior envelope leakage. In terms of energy efficiency and 

utility bill savings, exterior envelope leakage is the key. But, like guarded blower door testing, 

whole-building blower door tests require substantial resources in equipment, personnel, and time

(Hynek 2011).

The simpler and more common method for measuring air leakage in attached dwellings is to use 

a single blower door to pressurize and/or depressurize the test unit. This “single unit,” “total,” or 

“solo” (SO) test method measures the combination of air leakage between adjacent units through 

common surfaces as well as air leakage to the outside. Minimizing total leakage, or 

compartmentalization, is good practice to avoid indoor air quality issues between units, reduce 

energy use, reduce pressure differentials between units, and control stack effect. However, two 

significant limitations of the SO leakage test are:

• For retrofit work, if total leakage is assumed to be all to the outside, the energy benefits 

of air sealing can be significantly overpredicted. 

• For new construction, the total leakage values may result in failing to meet an energybased house tightness program criterion.

As described in a recently completed study by the Heschong Mahone Group (HMG 2012), the 

high cost and building occupant disruption of blower door testing deters utility program 

participants from pursuing building envelope improvements. 

“Until a protocol is established that provides a clear path with limited costs and risks for 

participation, building owners will continue to be hesitant to participate in whole building 

upgrade (retrofit) programs.”

In March, a Building America Expert Meeting was held to discuss the various methods, barriers 

to implementation, and current practices for measuring air changes and/or envelope leakage in 

attached dwellings. Attendees included representatives from federal government agencies, 

weatherization industry trainers and practitioners, building science researchers, and national 

laboratories. Presentations on multifamily dwelling air leakage were given by several building 

science researchers to encourage discussion on understanding why and how air leakage is 
2

measured, the challenges and costs associated with the measurements, and the need to conduct 

research to improve the process. 

A key objective of this meeting was to provide a forum for researchers and service providers to 

share information on their current practices and challenges and exchange ideas on opportunities 

for improvement. There was definitely a consensus for a simpler, but adequate, approach to 

assessing air leakage, both total and external.

A practical method needs to be developed to quantify the fraction of total leakage that is to the 

outside, and that is the objective of this research. With this simplification, measurement 

protocols can focus on the more practical single-dwelling unit total leakage test.
3

2 Research Objectives

The objective of the current research effort is to determine the viability of developing an 

algorithm that estimates the quantity of air leakage to the outside based upon a total leakage 

measurement and limited building and/or dwelling unit characteristic information. 

The primary questions to be addressed by this research are:

• Conceptually, how would a method for estimating outside air leakage from a total 

leakage measurement work? 

• Do preliminary analyses suggest that a simplified, reliable method can be developed?

• What additional research is needed to develop the concept into a viable, industryaccepted method?
4

3 Technical Approach

The theoretical concept for the simplified method is that a SO (total) leakage test result and a 

limited amount of building and/or apartment characteristic information can define a multiplier 

that provides the fraction of the SO (total) leakage that is to the outside.

The Expert Meeting in March was held to confirm interest and provide early guidance to the 

methodology development. Several meeting participants agreed to serve on an advisory panel,

and an initial meeting of the panel was held in conjunction with the summer ASHRAE meeting 

in San Antonio. Suggestions for approaching the analysis were provided. 

To demonstrate the viability of an outdoor-to-total leakage algorithm, analysis was performed on 

a limited set of data. Air leakage and building characteristic data were obtained from internal 

project resources at Steven Winter Associates, Inc. and Community Housing Partners’ New 

River Center for Energy Research and Training. Datasets include building characteristic 

information, dwelling unit information, and blower test results, both SO and guarded. 

FG and SO blower door test results were obtained from four multifamily low-rise projects. Table 

1 shows a summary description of the projects. While the projects are not geographically diverse, 

they do represent different types of attached housing and different ages.

Table 1. Summary of Building Used

Project Location Type of 

Building

Number of 

Buildings

Number of 

Units

Year of 

Construction

1 Winchester, 

Virginia

Garden-style 

apartments

2 14 1989

2 Newport News, 

Virginia

Garden-style 

apartments 5 35 1980

3 Roanoke, 

Virginia Rowhouses 4 22 1970

4 Staten Island, 

New York

Duplex 

rowhouses 4 41 2010

Projects 1 and 2 are both complexes of garden-style apartments in two- and three-story buildings.

Construction is wood frame on slab. Project 3 is a complex of 100 row houses with two to four 

bedrooms in 10 two-story wood frame-on-slab buildings. Project 4 is a newly constructed

ENERGY STAR® Home project using steel framing that presented air sealing challenges. 

Extensive blower door testing was done to achieve performance goals. These data were included 

in the analysis to provide a different type of construction. The two-family row houses or 

townhomes have a one-bedroom unit on the ground floor with a two-story, three-bedroom unit 

above.
5

4 Model Development Methodology

Three models were examined with primary emphasis on a multivariable linear regression 

(MVLR) model. Random Forest (RF) method and a simple ratio method of exposed surface area 

to total surface area (RESATSA) were the other models. The two models were studied with the 

purpose of confirming results from the MVLR model and exploring a simpler way of predicting 

leakage to outside from a SO test result. For all three models, the target response was the ratio of 

FG to SO test value. With this ratio, one can obtain the FG test value given the SO test value and 

vice versa. 

The following section describes each model and their results. The tool used for modeling was R. 

R is a language and environment for statistical computing and graphics (R Development Core 

Team 2008).

4.1 Multivariate Linear Regression Analysis

Statistical analyses were performed to create a model that would predict FG blower door or 

leakage to outside (LTO) values, from SO blower door results. An MVLR model was chosen:

Y = β + β X + β X + + β X + ε p ... 0 1 1 2 2 1

where

Υ = RFS = predicted ratio of FG to SO.

Xp = variables/predictors describing various building specifications, and p is the 

number of predictors being considered.

βp = partial correlation coefficient (PCC). It represents the change in RFS associated 

with a unit increase in the value of Xp (p

th variable) when all other variables are 

kept constant.

βo = intercept. Geometrically, it represents the value of RFS, where the regression 

surface (or plane) crosses the Y axis. Substantively, it is the expected value of RFS

when all the variable or predictors are equal to 0.

ε = the deviation of the value RFS from the mean value of the distribution given Xp.

This error term may be conceived as representing (1) the effects on RFS of 

variables not explicitly included in the equation, and (2) a residual random 

element in the dependent variable. The basic idea behind creating a linear 

accurate model is to minimize ε for each prediction.

Thirty-two variables were considered as possible candidates for creating the model. However, 

not all 32 variables could be used, since the goal of this research was to create a strong MVLR 

model with the smallest number of predictor variables that are also significant. Figure 1 shows a 

flowchart describing the procedure taken to identify the most significant predictors. The data 

sample used consisted of 112 observations containing 32 building specifications per apartment 

unit and measured RFS. 
6

Figure 1. Process for creating model

Backward 

Elimination 

Procedure

After all 

variables are 

tested then

Select valid 

variables/predictors based 

on perceived knowledge.

Create final model with 

most significant 

predictors

Create a list of all possible 

building specification 

variables 

YES

Remove variable 

from full model

NO

Compare R2 values of full 

model to reduced model 

using F-Test

Create full model with 

remaining predictors and 

reduced model with one less 

predictor (different one each 

time)

Is the reduced 

model more 

significant?
7

The model was created using an iterative process of identifying and eliminating less significant 

variables. A full model was created from all valid variables, and a reduced model was created 

using one less predictor. The F-test was then used to investigate the significance of the variable 

removed by comparing the R2 value of the full model to that of the reduced model. R2 is a 

measurement of how much variation in the data is explained by a model. An appropriate null 

hypothesis for the F-test is:

H0(β1=0) 2

against the alternative that the predictors β2...p ≠ 0. Thus, the reduced model in this case is:

RFS = β2X2+ ...+ βqXq 3

The F value is calculated as:

( ) ( )

( ) ( )

, 1 / 1

/

2

2 2

− − −

− − = R n p

R R p q F

p

p q d. f . = p − q, n − p −1 4

where

Rp = the sample multiple correlation coefficient that is obtained when the full model with all 

p variables, is fitted to the data.

Rq = the sample multiple correlation coefficient when the reduced model is fitted with q 

specific variables.

n = number of observations

p = number of variables in full model

q = number of variable in reduced model

The calculated or observed F value from Equation 4 is compared to tabulated F values with 

degrees of freedom of p-q and n-p-1 at a 95% confidence level. If the calculated F is greater than 

the tabulated F, the null hypothesis, Equation 2, is rejected. This means the coefficient β1 ≠ 0 and 

is therefore significant to the model. If not, the model is recreated without the variable in 

question, β1. 

The process is then repeated for all variables in the model. This method of iterating through 

equations and dropping least significant predictors is also called the backward elimination 

procedure. Obtaining a limited number of predictor variables helps isolate the most important 

variables and provide a simple and reusable model for builders, raters, engineers, and architects 

performing blower door tests. A final model was then created using the most significant 

predictors from the statistical analysis. 

In order to make the statistical analysis valid, the following necessary assumptions were made:
8

• Predictors are independent and normally distributed with mean zero and constant 

variance.

• Explanatory variables are nonstochastic: the values of the predictors were measured or 

obtained in advance.

• Predictors were measured without errors.

4.2 Predictor Variables Considered

A list of all variables considered can be found in Appendix A. In order to get a holistic sample of 

variables both quantitative and qualitative predictors were selected. Figure 2 shows an 

illustration of some of the variables considered for the model. 

Figure 2. Illustration of typical building specification

Quantitative predictors describe building specifications that can be quantified such as footprint 

area, total wall area, and exposed wall area. Qualitative predictors describe categorical building 

specifications like duct location, type of wall insulation, and unit location. For every categorical 

variable used, a set of dummy variables was created to describe each level in the variable by 

using a dummy coding scheme (UCLA: Statistical Consulting Group). This coding scheme 

assigns a value of 1 or 0 to the dummy variable by comparing each level of the categorical 

variable to a fixed reference level. The numerical values of the dummy variables are not intended 

to represent quantitative ordering, but only serve to identify the levels in a categorical variable

(Chatterjee and Bertram 1938). Table 2 shows an example of dummy variables and 

corresponding values created for Ductwork Location.
9

Table 2. Coding Scheme for Categorical Variables

Dummy 

Variables 

Number

Dummy Variables for Duct 

Location

Dummy_Var_1 

Versus 

Dummy_Var_2

Dummy_Var_1 

Versus 

Dummy_Var_3

1 Ductwork Location-None 0 0

2 Ductwork Location-Conditioned 

Space 1 0

3 Ductwork Location-Unconditioned 

Space 0 1

The fixed level in the dummy variable for Ductwork Location is “None” (Dummy_Var_1). This 

is case where a unit has no ducts. The reference point is seen as the point where all dummy 

variables are set to zero. As such, dummy variable Ductwork Location-None is represented when 

the two other Ductwork Location levels are set to zero. There were no units with ducts located in 

both conditioned and unconditioned space within the data analyzed. If there were such units, 

another level in the dummy variable would have been created for Ductwork LocationConditioned Space and Ductwork Location Unconditioned Space.

4.3 Major Predictive Variables

The Consortium for Advanced Residential Buildings (CARB) considered 32 variables, Appendix 

A, that are likely to affect infiltration values and tried to populate them for each unit given the

preliminary information received. Due to several variables having missing values and/or one

value, the full list was trimmed to the 13 variables shown in Table 3. Examples of variables with 

one value include window types were all double hung and building framing types were all 

platforms, thus two variables would not be beneficial to the model.

Table 3. Initial MVLR Model Results

Variable Number Variable

1 Ductwork location

2 Unit floor level

3 Unit location

4 Total exposed area

5 Window area

6 Exposed floor area

7 Exposed wall area

8 Footprint area

9 Total surface area

10 Ceiling area

11 Total shared surface area

12 Volume

13 Common wall area

Most of the variables shown in Table 3 are highly correlated with each other or collinear. For 

example, if footprint area is increased, the volume of the apartment unit also increases. This issue 
10

could cause redundant variables within a model. Thus, to reduce collinearity, new variables were 

created by creating ratios between variables. For example, the ratio of common wall to total 

surface area (RCWTSA) was created with the idea of capturing the relationship between 

common wall, standardized by total surface area, and RFS. Another ratio created was the ratio of 

window areas to total exposed surface area (RWATESA). Several ratios could be created, such 

as total exposed area to total surface area (RTESATSA), total surface area to volume (RTSAV), 

and exposed floor area to footprint area (REFAFPA). These ratios create more interactions

between variables in addition to the additive interaction facilitated in MVLR models. Different 

interactions between variables often improve the strength and accuracy of the model. By using 

the above-mentioned ratios, variables 4 to 13 from Table 3 were safely dropped. For the sake of 

having a concise description of the process of selecting the most significant variables, only two 

ratios along with the remaining valid categorical variables were selected. Future analyses will 

include more ratio variables and other interactions by multiplying, squaring, adding, or 

subtracting variables.

The remaining variables after elimination are:

1. RCWTSA, X1

1. RWATESA , X2

2. Unit Location

o End, X3

o Interior, X4

3. Duct Location

o Duct Location – Conditioned Space, X5

o Duct Location – Unconditioned Space, X6

o Duct Location – None, X7

4. Unit Level

o Bottom Floor, X8

o Top Floor, X9

o Middle Floor, X10

The first two predictors, RCWTSA (X1) and RWATESA (X2), are quantitative variables, whereas

Unit Location, Duct Location and Unit Level are qualitative/categorical variables. Ductwork

Location indicates whether the unit’s ductwork is in conditioned or unconditioned space or not 

present at all. Unit Location indicates whether a unit is an end or interior unit. Unit Level 

indicates whether a unit is on the top, bottom, or middle floor of an apartment building. Due to 

the small sample size, two-story units were described as top units in order to reduce the number 

of variable predictors used for such a small data sample. Future modeling with more data will 

certainly include another level in the dummy variable describing multistory units. As mentioned 
11

above, categorical variables are usually represented with dummy variables to denote each level 

in the variable. As such, Unit Location uses two dummy variables (X3 and X4) and Ductwork

Location (X5, X6 and X7) and Unit Level (X8, X9 and X10) use three dummy variables each, 

resulting in a total of ten predictors.

Thus, the MVLR equation becomes:

0 1 1 2 2 10 10 RFS = β +X β +X β + ... +X β 5

Knowing the predicted value of RFS from Equation 5, FG or LTO at CFM50 can be predicted 

using Equation 6, which multiplies predicted RFS by the measured SO blower door value:

FG = RFS × SO 6

4.4 Multivariable Linear Model Results

A summary of the MVLR model results is shown in Table 4. The Pr(>|t|) column is used to 

assess the significance of each variable. Pr(>|t|) is the probability that the PCC of a variable is

equal to zero, according to the t test. The t-test, is used in this analysis to determine the 

significance of a variable’s coefficient. For dummy variables, the Pr(>|t|) value is the probability

that the difference between the dummy variable and its reference point is zero.

As a general rule, a 5% or less probability is the acceptable benchmark to reject the null

hypothesis that a predictor variable’s PPC is zero; that is, the variable is insignificant. From 

Table 4, the probability that the difference between Ductwork Location-Conditioned Space and 

None is zero, as such there is no significant difference between the two Ductwork Location 

dummy variables. Since Ductwork Location is a variable that often affects infiltration values, 

CARB decided to leave it in the model; with a larger data sample the significance of dummy 

variable Ductwork Location will likely be realized. The intercept is the least significant 

nondummy predictor and it is not surprising that it has the lowest PCC and confidence level.

Insignificant variables will be dropped at this point, however, since this reports seeks to 

demonstrate more about the statistical process than the production of a finalized model, the 

intercept will not be discarded.

Table 4. MVLR Model Results

Predictors/Variables Estimate Pr(>|t|) Confidence 

Level

Intercept 0.179 0.373 < 90.0%

X1 RCWTSA 0.712 0.008 95.0%

X2 RWATESA –0.904 0.009 99.0%

X3 Unit Location-End 0.139 0.003 95.0%

X5

Ductwork LocationConditioned Space 0.026 0.704 < 90.0%

X6

Ductwork LocationUnconditioned Space 0.156 6.63e-08 99.9%

X8 Unit Floor Level: Bottom 0.304 1.20e-06 99.9%

X9 Unit Floor Level: Top 0.230 3.51e-06 99.9%
12

After evaluating the significance of each variable’s coefficient, the backward elimination 

procedure was performed to assess the significance of each variable, to the model. As discussed 

in Section 4.1, the F-test was done to determine the significance of the reduced model with 

respect to the full model at a 95% confidence level. The F-test is a measure of how relevant a 

variable is to the model (Chatterjee and Bertram 1938). The F-value calculated for each reduced

model was less than the corresponding tabulated F-values. As such, each null hypothesis was

rejected, implying that all variables in the full model are significant to the model. It must be 

noted that the significance of these predictors could change with a different sample data. 

The model was finally recreated and shown in Equation 7:

1 2 3 5 6 8 23 9 RFS = 0.18 + 0.71X −0.90X +0.14X − 0.03X +0.16X +0.30X +0. X 7

As mentioned above, dummy variables were created using a coding scheme, which chooses a 

level in the categorical variable as a reference point. The reference point is seen as the point 

when all dummy variables are set to zero. As such, dummy variable Unit Location-Interior (X4)

is represented when Unit Location-End (X3) is zero, Ductwork Location-None (X7) is 

represented when the two other Ductwork Location levels (X5 and X6) ) are set to zero, and Unit 

Floor Level is middle (X10), when Top (X9) and Bottom (X8) are set to zero. Consequently, 

depending on the values of the categorical variables for each input observation, the MVLR 

model will change as shown in Table 5. 

Table 5. MVLR Model Equations

Unit 

Location Duct Location Unit Floor Level

MVLR Model

(Unit and Duct Location), RFS

Interior (X4) None (X7) Middle (X10) 0.18 + 0.71X1 – 0.90X2

End (X3) None (X7) Middle (X10) 0.18 + 0.71X1 – 0.90X2 + 0.14X3 

Interior (X4) Ductwork LocationConditioned Space (X5) Middle (X10) 0.18 + 0.71X1 – 0. 90X2 – 0.03X5

End (X3) Ductwork LocationConditioned Space (X5) Middle (X10) 0.18 + 0.71X1 – 0. 90X2 + 0.14X3 –

0.03X5

Interior (X4) Ductwork LocationUnconditioned Space (X6) Middle (X10) 0.18 + 0.71X1 – 0.90X2 + 0.16X6

End (X3) Ductwork LocationUnconditioned Space (X6) Middle (X10) 0.18 + 0.71X1 – 0.90X2 + 0.14X3 +

0.16X6

Figure 3 shows the final model predicted values against measured values. The red diagonal line 

represents an accurate predictive model. The closer the data points are to the red line the more 

accurate the model, thus the MVLR model is fairly accurate.
13

Figure 3. Plot of MVLR model fitted against measured RFS

Statistical tests and confidence intervals used above are based on the assumption of normal 

errors. Figure 4 shows four plots that validate the assumptions made. The top left plot shows 

residuals against the fitted values with data points randomly distributed around the dotted

horizontal line at 0 and lie between ± 0.3 on the vertical axes. This shows that there is no 

discernible pattern to the distribution of residuals, supporting the assumption of normality in the 

data. The scale-location plot in the lower left shows the square root of the standardized residuals 

(square root of relative error) as a function of the fitted values. Again, for normally distributed 

data, there should be no obvious trend and that is also confirmed in this plot. The normal Q-Q is 

an informal graphical test of the hypothesis that a data sequence is normally distributed. If the 

plotted data points fall exactly on dotted diagonal line, the normal distribution assumption holds

(Khattree and Naik 1999). In this case, data points align fairly close to the diagonal line with a 

few outliers at the ends, thus the assumption of normality still stands. Finally, the plot in the 

lower right shows each data point’s leverage, which is a measure of its importance in 

determining the regression result. Superimposed on the plot are contour lines for Cook’s 

distance, another measure of the importance of each observation to the regression (Van Steen et 

al. 2001). Smaller distances mean that removing the observation has little effect on the regression 

results. Distances larger than 1, (not shown because of limited axis range) are suspicious and 

suggest the presence of possible outliers or a poor model. This MVLR model shows only three 

obvious outliers (94th, 95th, and 101st observations) out of 112 observations, as such the 

importance each observation is evenly spread.
14

Figure 4. Plots validating linear model
15

Figure 5. Plot of Rstudent residuals against predictors

Figure 5 displays the Rstudent residuals against each predictor variable. Rstudent residuals, also 

known as studentized residuals, is a standard way of measuring each predictor variable’s

contribution to the residuals of a model (Cook and Weisberg 1982). These graphs above are used 

to determine nonlinearity between a variable and predicted values, RFS, as well as outliers in data 

points. For quantitative variables RCWTSA ( X1) on the top left, RWATESA ( X2) on the top 

right, and fitted values on the bottom right, their data points are evenly distributed across the zero 

line and show no systematic pattern. As such, it is concluded that there is a linear relation

between selected predicted variables and the response, RFS. For categorical variables not much 

can be said about linearity, however, one can notice distinct outliers. These outliers could be the 
16

result of discrepancies in the model predictions. Removing these points will slightly improve the 

model; however, due to limitations in sample size, this research focuses more on the analytical 

methodology of arriving at an accurate and reusable model than on creating a finalized model.

Next CARB investigated the possibility of having a predicting model with other modeling types.

4.4.1 Random Forest Approach

Random Forest is a statistical modeling tool that implements Breiman’s random forest algorithm 

for classification and regression. 

It produces a regression tree which is built recursively from the data sample into 

more and more homogeneous groups, until a terminal node is reached. Each split 

is based on the values of one variable and is selected according to a splitting 

criterion. Once a tree has been built, the response for any observation can be 

predicted by following the path from the root node down to the appropriate 

terminal node of the tree. Based on the observed values for the splitting variables,

the predicted response value is simply the average response in that terminal node

(Grömping 2009).

As with other highly computational procedures, RF does not have a simple representation such 

as a formula (e.g., linear regression model) for the relationship between the predictor variables 

and the predicted values (Cutler et al. 2007). This makes interpretation and field application 

difficult. Appendix B shows an example of a regression tree that describes the RF model for this 

research. Random Forest was used in this research only to validate the results from the MVLR

model. For more details about this modeling tool see Liaw and Weiner (2002).

Figure 6. RF model results
17

A plot of predicted against measured RFS is shown in Figure 6, and once again a comparison of 

the data points with the red line gives a visual of the accuracy of the model. Since the data points 

are evenly spread and close to the diagonal red line, it is concluded that the RF model is also 

fairly accurate.

4.4.2 Area Weighted Approach

The area weighted approach has been suggested as a very simple method that may be good 

enough. The available sample data were tested against the validity of this approach. The method 

basically describes RFS as being equal to the ratio of exposed surface area, ESA, to total surface 

area, TSA, of the apartment envelope:

SO

FG

TSA

ESA RFS = = 8

ESA is the area of building including walls, windows, doors, and roofs in contact with ambient 

conditions. The reason behind exploring this model was to investigate a much simpler approach 

of predicting the RFS.

Predicted FG can then be calculated as:

SO

TSA

ESA FG = × 9

FG = RESATSA× SO 10

where

RESATSA = Ratio of exposed surface area to total surface area
18

Figure 7. RESATSA model results

Figure 7 shows the RESATSA model results by plotting predicted against measured RFS. The 

graph shows a wider spread in data points that are also relatively far from the red diagonal line. 

Thus, the predictive ability of this RESATSA model is limited. A detailed comparison of the 

three models’ results and their relative accuracy is described below.
19

5 Comparing Models

Each model, with the exception of RF, has a simple linear equation that can be used to calculate 

predicted FG by multiplying SO by the predicted RFS. There are several ways to measure quality 

of a model. In this research two parameters are used (Table 6).

Table 6. Summary of Regression Accuracy

Model Residual Standard 

Error

Adjusted

R2

RF 0.011 0.580

MVLR 0.109 0.553

RESATSA 0.249 0.065

Residual standard error explains the discrepancy between the measured and predicted RFS, taking 

into consideration the degrees of freedom, which is the difference between the numbers of 

observations in the data sample and the number of predictor variables, including the intercept. 

The lower the residual standard error is, the stronger the model. RF shows the lowest residual 

standard error, 0.011, meaning one should expect the smallest error between predicted and 

measured RFS when using RF model. This is followed by MVLR model and then RESATSA 

model.

Adjusted R2 is obtained from R2 value however, adjusted R2 accounts for the degrees of freedom 

in the model. The value of R2 ranges from 0 to 1 and the closer the value to 1 the more accurate 

the model is. A larger adjusted R2 value implies that a greater proportion of the variance in the 

data can be explained by the model. MVLR shows an adjusted R2 value, 0.553. Adjusted R2 of 

0.55, for a model created from measured values that have a wide range of uncertainly, is 

statically justifiable but certainly a higher R2 will be most preferred. Random Forest model had 

the highest R2 value, 0.58, but due to its high complexity, the model’s practicality in the field 

may not be feasible. With the lowest R2 value, REASTSA model comes out to be the worst, 

explaining only 6.5% of the variance in measured response.

The final MVLR model includes variables RCWTSA ( X1), RWATESA (X2), Unit Location-End

(X3), Unit Location-Interior(X4), Ductwork Location-Conditioned Space (X5), Ductwork

Location-Unconditioned Space (X6), Unit Floor Level-Top (X8) and Unit Floor Level-Bottom

(X9). Although the model’s variables and coefficients could change depending on size and 

variation in the sample dataset, the selected variables and coefficients, in this case are arguably 

credible. 

The predictability of this MVLR model is limited to the data sample and cannot be applied in all 

situations. Sample data in this research lack variety in terms of climate zone (only mixed-humid 

locations), foundation type (only slab-on-grade), and wall insulation (only dense pack and loose 

fiber glass), to mention a few. As such the model is expected to do poorly with data from other 

climate zones with different foundation and wall insulation types. More data with more 
20

variations would help obtain a stronger and more reliable model. There is also the possibility that 

the significance of predictor variables would change.

Moreover, a good variable selection procedure will result in several sets rather than a so called 

single “best” set of predictor variables (Chatterjee and Bertram 1938). With a larger, well-varied 

data sample, various sets of adequate equations would be obtained. The best choice of equation 

will boil down to which equation has the predictor variables easily obtained.

In this research, CARB basically demonstrated the possibility of having a correlation between

building geometric characteristics, location of duct, and the relationship of the dwelling to 

adjacent dwellings and RFS. This was achieved by creating three models with all 112 

observations and analyzing the correlation between select significant variables and the response, 

RFS. To varying extents, all three models prove a predictable relationship between building 

specifications and RFS. Considering the ease of usability in the field, the most practical model

was the MVLR model. The result of this research was certainly limited by the relatively small

and less varied sample data size. CARB’s goal is to create a model that has a strong predictive 

power in different climate zones, construction types, and building specifications, as such this 

model is still in development stages. The objective of this research was to demonstrate the 

MVLR model’s viability and development process.
21

6 Preliminary Test Case

A preliminary test of the MVLR model was run on blower door results for four attached 

rowhouses from the King County Housing Authority in Washington (Figure 8). 

Figure 8. Floor plan of King County Housing Authority apartment units

33 34

35 36
22

The data consisted of SO test results for each unit, pre- and post-retrofit, and a whole-building 

leakage test using multiple blower door systems. While we did not have SO and FG tests for 

each unit, we thought these data could still be used for demonstration purposes. Ideally, the sum 

of FG tests on each unit would be equivalent to the whole-building leakage test result. 

For this demonstration, we used the SO test data (left side of Table 7) to predict FG for each unit

(right side) using the preliminary model, Equation 7, and then summed the predicted FGs and 

compared the results to the measured whole-building leakage values. 

Table 7. MVLR Model Predictions Versus Sample Data

Unit No.

Measured SO 

(CFM 50) Predicted RFS

Predicted FG

(CFM 50)

PreRetrofitPostRetrofitPreRetrofitPostRetrofit

A 930 580 0.71 ± 0.07 660 ± 70 410 ± 40

B 900 720 0.78 ± 0.05 700 ± 40 560 ± 40

C 930 620 0.92 ± 0.09 850 ± 100 570 ± 60

D 900 590 0.85 ± 0.06 760 ± 50 500 ± 30

Measured LTO 

Whole Bldg. 

(All 4 Units)

3,340 1,930

Predicted FG 

Whole Bldg. 

(All 4 Units)

2,980 ± 260 2,040 ± 170

Measured post-FG for the whole building, 1,930 CFM, falls within the range of predicted FG 

2,040 ± 170 CFM, whereas for pre-infiltration, 3,340 CFM is just above the upper limit of the

predicted range of 2,980 ± 260. 

The result above is debatably a decent approximation, bearing in mind the limited dataset used to 

develop this model. Note the differences indicated between the pre- and post-retrofit SO and 

whole-building test results are not intuitive. It is not relevant to this exercise, but does suggest 

that there is uncertainty in blower door testing and its interpretation. The point of this exercise 

was to demonstrate the prediction methodology proposed.
23

7 Discussion

The analyses to date are encouraging, but given the very limited dataset studied, it is too early to 

draw conclusions. To apply the model to most any type of low-rise attached housing, the dataset 

from which the model is developed needs to include variety in age, construction materials, and 

configuration.

Our next step will be to identify and gather significantly more data from organizations that have 

conducted both SO and FG or whole-building testing on projects. This somewhat redundant 

testing is not commonly done, but we believe that research, weatherization training, and utility 

program design and evaluation firms may be sources. We also know of instances where SO tests 

were initially performed, but when compliance targets were not achieved, a whole-building test 

was performed.

Through continued research we will attempt to answer the following questions:

• Does the rigor of the predictive model hold up through analysis of more buildings, 

different types of construction, different climate regions, etc.? As the dataset becomes 

more varied, does it become more difficult to predict with acceptable accuracy the 

leakage to outside for a particular dwelling?

• Do the predictive variables change? Are more predictive variables needed to achieve 

acceptable accuracy? Is the model still simple enough to be useful? 

• Are different models necessary for predicting pre- and post-retrofit conditions? It’s quite 

possible that retrofit activity will address with some consistency certain sources of 

leakage. This possible bias might warrant a separate model or could be within the noise 

of a single model. If enough data can be collected for both pre- and post-retrofit testing, 

this will be examined.

• How good is good enough? A question that was discussed without resolution during the 

Expert Meeting is, How accurate does the estimate of outside air leakage need to be? The 

value is used for many purposes, including determining the cost effectiveness of air 

sealing and the adequacy of fresh air for ventilation. Perhaps the algorithm would be 

most appropriate as a screening tool. Outcomes would be yes, no, or maybe. The 

“maybe” outcomes would require more rigorous testing. CARB will seek input on this 

question from project advisors and efficiency program managers.
24

8 Conclusion

The research performed to date was successful with the following key findings:

• Conceptually, how would the new method work? The framework for a simple algorithm 

based upon a SO blower door test result and a few basic dwelling unit characteristics has 

been outlined (Section 4.4, Table 5).

• Do preliminary analyses suggest that a simplified, reliable method can be developed? 

Yes, while the subject dataset is very limited, preliminary analyses suggest that 

statistically significant predictors are present and can support the development of an 

algorithm.

• What additional research is needed to develop the concept into a viable, industryaccepted method? The next step is to collect additional data for analysis and algorithm 

development so they may be more broadly applied with confidence.
25

References

Chatterjee, S.; Bertram, P. (1938). Regression Analysis by Example. 2nd Edition. New York: John 

Wiley & Sons, Inc.

Cook, R. D.; Weisberg, S. (1982). Residuals and Influence in Regression. New York: Chapman 

and Hall.

Cutler, D.R.; Edwards, Jr., T.C.; Beard, K.H.; Cutler, A.; Hess, K.T.; Gibson, J.; Lawler, J.J.

(2007). “Random Forests for Classification in Ecology.” Ecology, 88(11):2783–2792.

Grömping, U. (2009). “Variable Importance Assessment in Regression: Linear Regression 

Versus Random Forest.” The American Statistician 63(4):308–319.

HMG. (2012). Study of Multi-Family Air Leakage Testing Strategies, Final Report, HMG Project 

Number 1014. Heschong Mahone Group, Inc., August 10, 2012.

Hynek, D. (2011). “Blower Door Testing in Multifamily Buildings,” Home Energy Magazine

September/October 2011. www.homeenergy.org/show/article/id/1711.

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Liaw, A.; Wiener, M. (2002). “Classification and Regression by randomForest.” R news 2(3):18–

22. Accessed November 26, 2012: www.webchem.science.ru.nl:8080/PRiNS/rF.pdf.

Lyons, J. (2013). Short-Term Test Results: Multifamily Home Deep Energy Efficiency Retrofit.

Golden, CO: NREL/SR-5500-56460. www.nrel.gov/docs/fy13osti/56460.pdf.

Neuhauser, K.; Bergey, D.; Osser, R. (2012). Leveraging Limited Scope for Maximum Benefit in 

Occupied Renovation of Uninsulated Cold Climate Multifamily Housing. 

http://apps1.eere.energy.gov/

buildings/publications/pdfs/building_america/renovation_uninsul_multifamily.pdf

R Development Core Team. (2008). R: A Language and Environment for Statistical Computing, 

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www.ats.ucla.edu/stat/r/modules/dummy_vars.htm. Accessed November 24, 2012.

Ueno, K.; Lstiburek, J.; Bergey, D. (2012). Multifamily Ventilation Retrofit Strategies. Golden, 

CO: National Renewable Energy Laboratory. NREL/SR-5500-56253. 

www.nrel.gov/docs/fy13osti/56253.pdf.

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buildings/publications/pdfs/building_america/byggmeister_test_home.pdf%20.
26

Appendix A: Predictor Variables Considered

Variable Description Format

Building and Unit Information

ApartmentComplex Name of apartment complex Text

BuildingNumber Building Number Text

BuildingType Building Type usu. a digit or letter 

(optional) Text

UnitConfiguration How are the unit set up in the apt 

complex Text

AgeOfBuilding Age of building of the apartment 

complex Number

Numberoffloors Number of floors in the apartment 

complex Number

Numberofunits Number of units in the apartment 

complex Number

Quantitative Specifications

FootPrintArea Foot Print Area of Unit Number

CeilingArea Ceiling Area of Unit Number

ExposedFloorArea Exposed flow area of Unit, ex 

cantilever Number

ExposedCeilingArea Exposed ceiling area usu. for a top 

floor Number

ExposedWallArea Exposed wall area i.e. wall in contact 

with outside Number

TotalExposedArea Same as wall area but includes 

ceiling for top unit Number

InteriorUnitHeight Height of Unit Number

Volume Volume of Unit Number

CommonWallArea Area of shared wall Number

TotalSharedSurfaceArea Total area of share surface include 

ceiling if there is a unit on top Number

WindowArea Total window area Number

NumberofPanes Number of panes in the windows Number

TotalWallArea Total wall area Number

DoorArea Total door area Number
27

Qualitative Specifications

Unit Story Level Top, Bottom or Middle Top, Bottom or Middle

Unit Location End, Interior End, Interior

FoundationType Type of foundation

Full Basement

Crawl space

Slab-on-Grade

ContructionMaterial Material used for the framing of 

the building

Wood

Steel Frame

Masonry

BuildingFramingType Building framing type Balloon

Platform

WallInsulation Type of all insulation used

Blanket: batts and rolls

Foam board

Insulating concrete forms 

(ICFs)

Loose-fill cellulose

Loose-fill fiber glass

Reflective/Bubble

Rigid fibrous

Sprayed foam

Structural insulated panels 

(SIPs)

Dense Pack

RoofInsulation Type of roof insulation used Same as WallInsulation

WindowType Type of window is used

Single Hung

Double Hung

Casement

Bay/Bow

Awning

Slider

WindowFrame Window framing material

Metal

Vinyl

Fiberglass

Wood

Siding Wall siding material

Stucco 

Vinyl 

Aluminum 

Wood 

Stone

Brick

Mixed

CommonWallConstruction Common wall construction 

material

Shaft Wall (2hr)

Double Wall (2hr)
28

Double Layer Drywall (1hr)

Concrete/Brick (2hr)

Heating Space heating system

Atmospheric

Sealed Combustion

Resistance

Cooling Space cooling distribution system Ducted

Non-ducted

DuctworkLocation Location of duct work Conditioned Space

Unconditioned Space
29

Appendix B: Random Forest Output
DOE/GO-102013-3939 ▪ July 2013

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