The cores that agree for the wrong reason

The strength three cores revise took three cores from an existing structure, mean 33 N/mm² and 12 per cent scatter, and added them to what the structure was specified to be: a grade whose concretes average 38 N/mm² with 15 per cent scatter, weighted as three specimens on its mean and ten degrees of freedom on its scatter. Together they gave a characteristic strength of 26.5 N/mm², which three cores alone reach only at twenty-one. The same arithmetic caught a concrete much weaker than its specification, because cores that disagree with the prior widen the scatter the update ends with.

Every core in that calculation was an independent draw from the structure’s concrete. Cores are rarely drilled that way. An assessment is usually given one bay to work in — the bay with the access, the bay that is being altered, the bay where the crack is — and three cores a metre apart in one slab share a pour, a curing, a day’s weather and the same delivery’s water content. They agree with each other more than three cores from three floors would, for reasons that have nothing to do with how good the building’s concrete is. The question that essay left was whether such a set should be counted as fewer cores, by how much, and whether the prior’s protection against a weak concrete survives once the cores agree for the wrong reason.

Two kinds of scatter in one building #

The model is the simplest one that holds the distinction. The logarithm of strength at a point in a structure is the structure’s mean, plus a bay effect shared by every point in that bay, plus a point effect of its own:

$$y = \mu + b_j + e, \qquad b_j \sim N(0,\ \rho\sigma^2), \qquad e \sim N(0,\ (1-\rho)\sigma^2).$$

Here $\sigma^2$ is the scatter of the structure’s concrete taken as a whole — what a characteristic value is a fractile of — and ρ is the share of it that lives between bays. Two cores from one bay share their $b_j$, so ρ is also the correlation between them. At ρ = 0 every core is independent of every other, wherever it was drilled. At ρ = 0.5 half the building’s variance is between pours and half within them. At ρ = 0.9 the bays differ a great deal and a bay is nearly uniform inside itself.

None of these is exotic. The Joint Committee on Structural Safety’s probabilistic model code describes the correlation of concrete strength between two points of one member as falling from one towards a half over a few metres, which puts two cores a metre apart in one slab near the top of that range. What the number is for a given building is not known before the cores are drilled, and three cores from one bay cannot measure it, because measuring a between-bay variance takes more than one bay.

What a bay of cores is worth #

The information a set of correlated cores carries about the structure’s mean has a clean form. With the cores’ correlation matrix $R$ known up to $\sigma^2$, the best estimate of μ is the generalised least-squares mean $\bar y = \mathbf 1^{\mathsf T} R^{-1} y / \mathbf 1^{\mathsf T} R^{-1} \mathbf 1$, and its variance is that of a mean of

$$n_{\text{eff}} = \mathbf 1^{\mathsf T} R^{-1} \mathbf 1$$

independent specimens. For $m$ cores from one bay, all correlated at ρ, that is $m/(1 + (m-1)\rho)$.

The curves fall fast. At a correlation of 0.2, three cores from one bay are worth 2.1 independent ones and twelve are worth 3.8. At 0.5, three are worth 1.5 and twelve only 1.8. At 0.8 the count hardly matters: three, six and twelve are each worth 1.2. The dotted ceiling is the reason. However many cores are drilled from one bay, they cannot be worth more than 1/ρ, because the bay effect they share is one draw, and no amount of drilling inside the bay reduces the uncertainty in it.

A bay is a specimen. Its cores measure the bay very well and the building only as well as one bay measures it, and the twelfth core from one slab adds almost nothing that the third had not.

The cores’ scatter is affected the other way. The spread of three cores about their own mean measures only the point effects, the $e$ in the model, whose variance is $(1-\rho)\sigma^2$. So three cores that agree closely are telling the truth about the bay and understating the building’s scatter by the factor 1 − ρ — a tenth of it at ρ = 0.9. The quantity with the right expectation is the sum of squares about $\bar y$ weighted by $R^{-1}$, $S = (y - \bar y)^{\mathsf T} R^{-1}(y - \bar y)$, which for a single bay is the ordinary sum of squares divided by 1 − ρ and is $\sigma^2$ times a $\chi^2$ on $n - 1$ degrees of freedom.

Both corrections drop straight into the conjugate update the earlier essay used. The cores’ weight on the mean becomes $n_{\text{eff}}$ instead of $n$, and their contribution to the scatter becomes $S$ instead of $(n-1)s^2$:

$$n'' = n' + n_{\text{eff}}, \qquad m'' = \frac{n' m' + n_{\text{eff}}\,\bar y}{n''}, \qquad \nu'' s''^2 = \nu' s'^2 + S + \frac{n' n_{\text{eff}}}{n''}\,(\bar y - m')^2,$$

with ν″ = ν′ + n as before. Counting the cores as independent is the same update with $R$ the identity. Everything that follows compares the two on a structure whose concrete is known, by drawing its bays and its cores two thousand times and asking, for each drilling, what share of the building’s concrete falls below the characteristic value that drilling produced. An honest 5 per cent characteristic value leaves 5 per cent below it on average.

Alone, counting them as three is the whole error #

The first case is the one the earlier essay began with: cores and nothing else.

Counted as independent, three cores from one bay produce a characteristic value that leaves 10.3 per cent of the structure’s concrete below it at a correlation of 0.5, and 26.5 per cent at 0.9. The value promised a twentieth and delivered a quarter. Both corrections push the same way here: the cores’ close agreement is read as a uniform concrete, and their mean is read as a well-determined mean, when it is one bay’s.

Counted by their correlation, the same three cores leave 4.9 and 5.0 per cent. The estimate is honest at every correlation, and it is honest because it is much lower. A drilling of 31, 33 and 35 N/mm² gives a characteristic value of 26.9 counted as independent; at ρ = 0.9 the same three numbers give 15.1, because their spread of 6 per cent is now understood to be a tenth of the building’s variance and their mean is worth a single specimen. That is the arithmetic’s correct reaction to three cores from one place: it knows almost nothing.

The comparison with the specimens a laboratory tests is exact. The strength that belongs to the test programme treated cubes as draws from a pour, and a set of cubes from one batch is the same kind of set as cores from one bay — informative about the batch and not about the supply. Conformity testing for new concrete specifies sampling across deliveries for exactly this reason. Assessment of old concrete inherits the statistics of conformity testing and very often not its sampling.

With the specification, the honest count is no safer #

The surprise comes when the prior returns.

The specification steadies both lines. Counted as independent, three cores from one bay now leave 6.2 per cent of the concrete below the estimate at a correlation of 0.5 and 7.4 per cent at 0.9 — the prior’s ten degrees of freedom on the scatter stop three tightly grouped cores from persuading the update that the building is uniform. That is the protection the earlier essay found, and it survives.

But the honest count does not improve on it. At 0.5 it leaves 6.5 per cent, slightly worse; at 0.9, 7.3, no better. The reason is in the update’s first line. Counting the cores as worth 1.5 or 1.1 specimens instead of three moves weight from the cores to the specification, and the specification is not neutral: it describes a population averaging 38 N/mm², and this building’s concrete averages 33. The cores were the only thing in the calculation saying the building is weaker than its grade, and the honest count turns their voice down. What the update gains in honesty about the scatter it loses in deference to a prior that is optimistic about the mean.

The light line is the same three cores drilled from three bays, one each. Uncorrelated draws need no correction, carry their full weight, and leave 4.8 per cent of the concrete below the estimate at every correlation the figure draws. Nothing in the arithmetic achieves what the drilling plan achieves for free.

A weak concrete hides behind its grade #

The earlier essay’s hardest test was a concrete 15 per cent weaker than its cores’ expectation — a mean of 28.1 N/mm², a characteristic of 22.9, below the grade it was ordered at. Three independent cores caught it: the conflict between their mean and the specification’s was added to the scatter, and the estimate came out within 0.4 N/mm² of the truth.

From one bay the protection fails, and it fails worst when the cores are counted honestly: 18.7 per cent of the concrete below the estimate at a correlation of 0.5, and 24.2 per cent at 0.9, against 11.5 and 13.6 counted as independent. The conflict term is the culprit, and its form says why. It is

$$\frac{n' n_{\text{eff}}}{n' + n_{\text{eff}}}\,(\bar y - m')^2,$$

and it is weighted by the cores’ worth. Three independent cores that disagree with the specification by 15 per cent contribute a conflict weighted by 1.5; the same three from one bay at ρ = 0.9 contribute one weighted by 0.8. The update has been told, correctly, that one bay’s disagreement with the specification could be one bad pour, and it duly discounts it. A drilling of 27, 28 and 29.3 N/mm² — a plainly weak bay — gives 23.7 counted as independent and 25.5 counted by its correlation, against a truth of 22.9.

What protected the earlier estimate was not the prior’s honesty; it was the cores’ independence. Three cores that disagree with a specification because the building is weak, and three cores that disagree because their bay is weak, look identical from inside the bay, and a careful update must treat them as the second.

The light line again does what the dashed and solid ones cannot. Three cores from three bays leave 8.7 per cent below — still above 5, as three independent cores did in the earlier essay, because three specimens against a confident prior can only pull so far, but flat across every correlation. Three bays that all disagree with the specification are three pieces of evidence, and the update treats them as three.

What one bay’s cores agree about #

The averages hide how the estimates are distributed, and the distribution is where the bay effect shows itself.

From one bay, the estimates average 27.2 N/mm² with a standard deviation of 1.2, and 62 drillings in a hundred produce a characteristic value above the structure’s own 26.9. From three bays they average 26.5 with a standard deviation of 1.3, and 38 in a hundred land above it. The two histograms are nearly the same width. The one-bay estimates are not noisier; they are displaced, towards the specification’s optimism, because every one of them gave the cores’ mean less weight than three independent cores would have earned.

That is the shape of the danger in an assessment. A set of three cores from one bay that agree closely and sit near the grade looks like a good result, and it is also exactly what a bay of good concrete in a mediocre building produces. The agreement is real. It is agreement about the bay.

Six cores, and where to put them #

The practical question is a drilling plan, and the figure that answers it holds the number of cores fixed.

Six cores from one bay are worth 1.1 independent ones, and leave 8.6 per cent of the concrete below the estimate. Split two ways they are worth 2.1 and leave 7.0; three ways, 3.2 and 6.0; six ways, 6.0 and 4.3 — below the target, and so slightly conservative, because six independent cores and a prior are a strong combination. The plan moves the result by more than every correction the arithmetic can make, and it costs only the walk between bays. In the earlier essay three cores with the specification were worth twenty-one cores alone; here, at a correlation of 0.9, six cores from one bay are worth fewer than two cores from two.

There is a cost, and it is not small for every structure. A core from a bay that is not being altered is a hole in a member that is in service, and the core drilled after the slab was built found that the place a core is easiest to take, beside a column, is the place the slab can least spare. But a core’s information is mostly its bay’s, and the second and third cores from one bay are the cheapest to drill and the least worth having. Spreading them is the economy.

By hand, for one drilling #

Take the drilling of 31, 33 and 35 N/mm². The logarithms are 3.434, 3.497 and 3.555, with mean 3.495 and sum of squares about it 0.00737.

Counted as independent and alone, $s^2 = 0.00737/2 = 0.00369$, $s = 0.061$, and with $t_{0.95,2} = 2.920$ and $\sqrt{1 + 1/3} = 1.155$, the characteristic value is $e^{3.495 - 2.920 \times 0.061 \times 1.155} = e^{3.290} = 26.9$ N/mm².

At ρ = 0.9 the cores are worth $3/(1 + 2 \times 0.9) = 1.07$ specimens and their scatter is $S = 0.00737/0.1 = 0.0737$ on two degrees of freedom, so $s = \sqrt{0.0369} = 0.192$. The multiplier is $2.920 \times \sqrt{1 + 1/1.07} = 4.06$, and the characteristic value is $e^{3.495 - 4.06 \times 0.192} = e^{2.716} = 15.1$ N/mm². The mean has not moved; everything else has.

With the specification as a prior, $n'' = 3 + 1.07 = 4.07$ and the revised log-mean is $(3 \times 3.627 + 1.07 \times 3.495)/4.07 = 3.592$, a median of 36.3 N/mm² — pulled most of the way back to the specification’s 37.6, where three independent cores had left it at 35.2. That single line is the whole of the weak-concrete failure: the cores’ mean is now a minority voice in its own update.

What the correlation between cores rests on #

The correlation is assumed known. It is not, and three cores from one bay cannot estimate it. Every result here uses the true correlation in the update, which is the most favourable case for the honest count, and an update that guesses ρ wrong lands between the two lines. The direction of the conclusion does not depend on the guess: from one bay, nothing in the update can tell a weak building from a weak bay.

The bays are independent of each other. A building’s floors were cast in sequence from one supplier, and adjacent bays may be closer to each other than to distant ones, as the floors that fill up together found for loads that share a cause. A spatial correlation that decays with distance would make two adjacent bays worth less than two distant ones, and the drilling plan’s advice becomes: spread across floors and pours, not across a corridor.

The structure is the target. If the question is the strength of one member and the cores come from that member, the bay effect is part of what is wanted, not a nuisance, and the cores are independent draws from it. Correlation only costs information when the cores are asked about concrete they did not come from. And the strength no specimen had still applies on top: a core’s strength is not a member’s.

The same fault in every pooled estimate #

The defect is not peculiar to concrete. The visits that come to every floor at once found the same structure in imposed loads: a reduction for many floors is earned only if their peaks are independent, and a correlation between floors takes back exactly what the independence gave. The load that is never all there at once priced the same assumption as a fractile. In each case a sum of $n$ things is credited with the averaging of $n$ independent draws, and in each case the honest count is $n/(1 + (n-1)\rho)$, with its ceiling of 1/ρ.

What is particular to an assessment is the third ingredient, the prior. A prior and a correlated sample interact in a way neither does alone: correlation tells the update to listen less to the data, and the update can only listen more to the prior instead. When the prior is a specification and the question is whether the building met it, that is the one direction the evidence cannot afford to be discounted in. The strength it had on the day found that a concrete’s strength is a history as much as a mix; a bay’s cores record that bay’s history, and only a spread of bays records the building’s.

Still open: cores that disagree about the bay effect #

Every drilling here takes cores from bays chosen at random and treats the correlation as known. A real assessment drilling two cores from each of four bays can estimate the correlation from its own data — the between-bay variance from the bays’ means, the within-bay variance from the pairs — and then update with the correlation it has measured. With so few bays that estimate is itself poor, and a hierarchical prior on the two variances, the between-pour scatter of the grade and the within-pour scatter of a member, would carry it the way the specification carried the mean. Whether two cores from each of four bays, with a prior on how a grade’s scatter divides between pours and points, can find both the building’s characteristic strength and the share of its scatter that lives between bays — or whether eight cores are still too few to separate the two, so that the plan, rather than the analysis, must keep deciding — is the question a drilling plan with pairs in it puts to this one.