Case Study

Case Study – Brick Wall Spalling

Claims

  • The owner claims that bricks were manufactured defectively.
  • The brick manufacturer counters that poor design and shoddy management led to the damage.

Goal

  • Determine the spall rate (damage rate per 1,000 bricks).

Experiment

  • The owner uses several scaffold-drop surveys, this survey type provides the most accurate estimate of spall rates in each wall segment.
  • Drops were made in areas with high spall concentrations, but not all damaged bricks could be made out from the photos.
  • The brick manufacturer divides the walls of the complex into 83 wall segments and takes a photo of each.
  • The number of damaged bricks for all 83 photos yields the total spall damage.

Construct a scatter diagram of data

The data shows how many bricks out of 1,000 were damaged at 11 drop locations for the two different experiments, “drop spall rate” and “photo spall rate”. The data shows an increasing trend from location 1 to location 11 (shown by using a least-squares prediction equation and the parameter B1 (the slope) is positive in both experiments).

Find the prediction equation for the drop spall rate using MINITAB

Regression Analysis: drop location versus photo spall rate (for 1,000 bricks)

The regression equation: drop location = 3.70 + 0.477*photo spall rate

(S= 1.73107, R-Sq = 75.5%, R-Sq(adj) = 72.8%)

Regression Analysis: drop location versus drop spall rate (for 1,000 bricks)

The regression equation: drop location = 3.28 + 0.172*drop spall rate

The elements of the designed experiment are the response variable (dependent variable) which is the spalling rate, and the factors (independent variable that possibly impacts the response variable) which is the type of experiment: ‘drop spalling” or “photo spalling”. The 11 locations are the levels for both factors. We could use these two experiments as our treatments (factor-level combinations) and conduct a completely randomized design, however, we must be careful doing so.

Earlier, we noticed an increasing amount of spalling (positive slope in the least-squares equation) between locations 1 and 11. This is problematic and means independent random samples were not selected for each treatment.

As a result of independent random samples not being selected for each treatment, we conduct a randomized block design. This better controls the sampling variability within the treatments (as measured by MSE). The randomized block design utilizes experimental units that are matched sets, assigning one from each set to each treatment. These matched sets, or blocks, allow ‘k’ experimental units (where ‘k’ is the number of treatments), that are as similar as possible to be grouped.

By employing blocks, the sampling variability of the experimental units in each block will be reduced, in turn reducing the measure of error, MSE. This tends to prevent a Type II error; to not reject the null hypothesis that the treatment means are equal when they differ. The conclusion that the treatments mean “drop spalling” and “photo spalling” are the same could be caused by not using blocks which makes the MSE very large. This faulty conclusion could be a function of how we designed our experiment.

There will be 11 blocks (for the 11 locations), 2 treatments (for the two experiments), and a total of 22 responses (n=bk).

Completely Randomized Design

  • One-way ANOVA (for 1,000 bricks): drop spall rate and photo spall rate
  • Randomized Block Design
  • Regression Analysis: drop location versus drop spall rate, photo spall rate, block mean
  • *The block mean is highly correlated with other X variables
  • *The block mean is removed from the equation

The regression equation (for 1,000 bricks): drop location = 3.31 + 0.152*drop spall rate + 0.057*photo spall rate

Conduct a formal statistical hypothesis test to determine if the photo spall rates contribute information for the prediction of drop spall rates

  • ANOVA F-Test to Compare ‘k’ Treatment Means: Randomized Block Design
  • Ho: µ1= µ2
  • Ha: At least two treatment means differ
  • Test statistic: F = MST/MSE
  • Rejection region: F > Fα, where Fα is based on (k- 1) numerator DOF & (n – b – k + 1) denominator DOF

Conditions Required for a Valid ANOVA F-Test: Randomized Block Design

  • The ‘b’ blocks are randomly selected, and all ‘k’ treatments are applied (in random order) to each block. Good.
  • The distributions of observations corresponding to all ‘bk’ block – treatment combinations are approximately normal. Good.
  • T-test performed since sample size n is small and has the effect of making the average spalling rate for each treatment deviate from the normal distribution.
  • The ‘bk’ block – treatment distributions have equal variances. Good.
  • (0.7566 = 0.1368 = 0.3879). Let’s assume we want the significance level to be 95% (α = 0.05).
  • From Table IX of Appendix A, with ν1 = 1 DOF & ν2 = 10 DOF, we find that F* = 4.96.

The F-ratio for the completely randomized design (factor – the type of experiment) = 4.06 is < tabled value = 4.96, so we do not reject the null hypothesis and assume the two treatments are equal. We could have arrived at the same conclusion by using the fact that the p-value = .058 is > α = .05. At this point, there is evidence we should employ a randomized block design.

The F-ratio for the randomized block design (factor – the type of experiment) = 14.85 is > tabled value = 4.96, so we accept the alternative hypothesis and assume the two treatments differ. We could have arrived at the same conclusion by using the fact that the p-value = 0.002 is < α = 0.05.

Comment on your findings

We should assume that the two tests yield different results. Therefore, we cannot determine the accurate rate of spalling, which was the goal from the onset.

From the randomized block design MINITAB results: S = 1.70833, R-Sq = 78.8%, R-Sq (adj) = 73.5%

This shows a fair amount of correlation between the two types of tests, r = 73.5, but usually, we are looking for the correlation coefficient to be 80% or higher (there are many instances where r is in the high 90s). The standard deviation for how many damaged bricks there are, for the randomized block design, is < 2 bricks per 1,000 bricks investigated. It seems very low compared to almost 13 for the completely randomized design. We could use the regression model to find the total amount of bricks damaged for the five-building apartment complex, but this seems risky since we don’t know if the spalling rate will continue to increase at approximately the same rate to the 83rd wall segment. A higher-order model or interaction model will not help the uncertainty of our sample of 11 walls not necessarily being representative of the whole population of 83 wall segments.